Ludwig Boltzmann
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Ludwig Eduard Boltzmann (Vienna, Austria-Hungary, February 20, 1844 – Duino near Trieste, September 5, 1906) was an Austrian physicist famous for the invention of statistical mechanics.
Boltzmann's most important scientific contributions were in kinetic theory, including the Maxwell-Boltzmann distribution for molecular speeds in a gas. In addition, Maxwell-Boltzmann statistics and the Boltzmann distribution over energies remain the foundations of classical statistical mechanics. They are applicable to the many phenomena that do not require quantum statistics and provide a remarkable insight into the meaning of temperature.
To quote Planck, The logarithmic connection between entropy and probability was first stated by L. Boltzmann in his kinetic theory of gases.Template:Ref This famous formula for entropy <math>S</math> isTemplate:Ref Template:Ref
- <math> S = k \; \log W</math>
where <math>k</math> = 1.3806505(24) × 10−23 J K−1 is Boltzmann's constant, and the logarithm is taken to the natural base <math>e</math>. <math>W</math> is the number of possible microstates corresponding to the macroscopic state of a system — the number of (unobservable) "ways" the (observable) thermodynamic state of a system can be realized by assigning different positions and momenta to the various molecules. Boltzmann’s paradigm was an ideal gas of <math>N</math> identical particles, of which <math>N_i</math> are in the <math>i</math>-th microscopic condition (range) of position and momentum. <math>W</math> can be counted using the formula for permutations
- <math> W = N!\; / \; \prod_i N_i! </math>
where i ranges over all possible molecular conditions. (<math>!</math> denotes factorial.) The "correction" in the denominator is due to the fact that identical particles in the same condition are indistinguishable. <math>W</math> is called the "thermodynamic probability" since it is an integer greater than one, while mathematical probabilities are always numbers between zero and one.
Much of the physics establishment rejected his thesis about the reality of atoms and molecules — a belief shared, however, by Maxwell in Scotland and Gibbs in the United States; and by most chemists since the discoveries of John Dalton in 1808. He had a long-running dispute with the editor of the preeminent German physics journal of his day, who refused to let Boltzmann refer to atoms and molecules as anything other than convenient theoretical constructs. Only a couple of years after Boltzmann's death, Perrin's studies of colloidal suspensions (1908-1909) confirmed the values of Avogadro's number and Boltzmann's constant, and convinced the world that the tiny particles really exist.
The equation for <math> S </math> is engraved on Boltzmann's tombstone at the Vienna Zentralfriedhof — his second grave.
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The Boltzmann equation
Image:Ludwig Boltzmann at U Vienna.JPG Template:Main
The Boltzmann equation was developed to describe the dynamics of an ideal gas.
- <math> \frac{\partial f}{\partial t}+ v \frac{\partial f}{\partial x}+ \frac{F}{m} \frac{\partial f}{\partial v} = \frac{\partial f}{\partial t}\left.{\!\!\frac{}{}}\right|_\mathrm{collision} </math>
where <math>f</math> represents the distribution function of single-particle position and momentum at a given time (see the Maxwell-Boltzmann distribution), <math>F</math> is a force, <math>m</math> is the mass of a particle, <math>t</math> is the time and <math>v</math> is an average velocity of particles.
This equation describes the temporal and spatial variation of the probability distribution for the position and momentum of a density distribution of a cloud of points in single-particle phase space. (See Hamiltonian mechanics.) The first term on the left-hand side represents the explicit time variation of the distribution function, while the second term gives the spatial variation, and the third term describes the effect of any force acting on the particles. The right-hand side of the equation represents the effect of collisions.
Image:Zentralfriedhof Vienna - Boltzmann.JPG
In principle, the above equation completely describes the dynamics of an ensemble of gas particles, given appropriate boundary conditions. This first-order differential equation has a deceptively simple appearance, since <math> f </math> can represent an arbitrary single-particle distribution function. Also, the force acting on the particles depends directly on the velocity distribution function f. The Boltzmann equation is notoriously difficult to integrate. David Hilbert spent years trying to solve it without any real success.
The form of the collision term assumed by Boltzmann was approximate. However for an ideal gas the standard Chapman-Enskog solution of the Boltzmann equation is highly accurate. It is only expected to lead to incorrect results for an ideal gas under shock-wave conditions.
Boltzmann tried for many years to "prove" the second law of thermodynamics using his gas-dynamical equation — his famous H-theorem.Template:Ref However the key assumption he made in formulating the collision term was "molecular chaos", an assumption which breaks time-reversal symmetry as is necessary for anything which could imply the second law. It was from the probabilistic assumption alone that Boltzmann's apparent success emanated, so his long dispute with Loschmidt and others over Loschmidt's paradox ultimately ended in his failure.
For higher density fluids, the Enskog equation was proposed. For moderately dense gases this equation, which reduces to the Boltzmann equation at low densities, is fairly accurate. However the Enskog equation is basically an heuristic approximation without any rigorous mathematical basis for extrapolating from low to higher densities.
Finally, in the 1970s E.G.D. Cohen and J.R. Dorfman proved that a systematic (power series) extension of the Boltzmann equation to high densities is mathematically impossible. Consequently nonequilibrium statistical mechanics for dense gases and liquids focuses on the Green-Kubo relations, the fluctuation theorem, and other approaches instead.
Energetics of evolution
In 1922, Alfred J. Lotka [1] referred to Boltzmann as one of the first proponents of the proposition that available energy (also called exergy) can be understood as the fundamental object under contention in the biological, or life-struggle and therefore also in the evolution of the organic world. Lotka interpreted Boltzmann's view to imply that available energy could be the central concept that unified physics and biology as a quantitative physical principle of evolution. Howard T. Odum later developed this view as the maximum power principle.
Significant contributions
1872 - Boltzmann equation; H-theorem
1877 - Boltzmann distribution; relationship between thermodynamic entropy and probability.
1884 - Derivation of the Stefan-Boltzmann law
Evaluations
Closely associated with a particular interpretation of the second law of thermodynamics, he is also credited in some quarters with anticipating quantum mechanics.
For detailed and technically informed account of Boltzmann's contributions to statistical mechanics consult the article by E.G.D. Cohen.
See also: Philosophy of thermal and statistical physics.
Final years
It appears that Boltzmann may have suffered from bipolar disorder. In 1906 he sadly committed suicide by hanging himself, while on holiday in Duino near Trieste.
Notes
- Template:Note 1. Max Planck, p. 119.
- Template:Note 2. The concept of entropy was introduced by Rudolf Clausius in 1865. He was the first to enunciate the second law of thermodynamics by saying that entropy always increases.
- Template:Note 3. An alternative is the information entropy definition introduced in 1948 by Claude Shannon.[2] It was intended for use in communication theory, but is applicable in all areas. It reduces to Boltzmann's expression when all the probabilities are equal, but can, of course, be used when they are not. Its virtue is that it yields immediate results without resorting to factorials or Stirling's approximation. Similar formulas are found, however, as far back as the work of Boltzmann, and explicitly in Gibbs (see reference).
- Template:Note 4. Wallace Carothers, who discovered neoprene and nylon and founded the science of long-chain polymers, finally drank his cocktail of cyanide-laced lemon juice in 1937, one year before nylon reached the market.
- Template:Note 5. See Tolman, Chapter VI, for an extensive discussion.
See also
External links
- Template:MacTutor Biography
- Scienceworld biography
- Biography from St Andrews University
- Boltzmann's Work in Statistical Physics Article by Jos Uffink for the Stanford Encyclopedia of Philosophy
- Claude E. Shannon, "The Mathematical Theory of Communication"
- Ludwig Boltzmann's Gravesite
References
- Template:Cite book English translation by Morton Masius of the 2nd ed. of Waermestrahlung. Reprinted by Dover (1959) & (1991). ISBN 0486668118
- Template:Cite book Reprinted: Dover (1979). ISBN 0486638960
- Template:Cite book
- David Lindley (Physicist) Boltzmann's Atom: The Great Debate That Launched A Revolution In Physics. ISBN 0684851865
- A.J.Lotka (1922) 'Contribution to the energetics of evolution'. Proc Natl Acad Sci, 8: pp. 147–51.bn:লুটউইখ বোল্ট্স্মান
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