October 29, 2014
Thermodynamics has developed from contributions from scientists of many nationalities. As just a few examples, we have Sadi Carno (1796-1832) from France; Robert Boyle (1627-1691) and William Thomson, Lord Kelvin (1824-1907) from Ireland; James Prescott Joule (1818-1889) from England; Rudolf Clausius (1822-1888) from Germany; James Clerk Maxwell (1831-1879) from Scotland; Josiah Willard Gibbs (1839-1903) from the United States; and Ludwig Boltzmann (1844-1906) from Austria).
It's only fitting that Greece, the origin of the Western intellectual tradition of philosophy and science, should also have been the birthplace of some prominent thermodynamicists. One of these is Constantin Carathéodory (1873-1950), whose profession was primarily mathematics.
In 1909, Carathéodory ventured into applied mathematics by working to formulate thermodynamics axiomatically. His book, "Investigations on the Foundations of Thermodynamics," sought to derive thermodynamics from mechanics. The importance of this approach is that he was able address irreversibility. Classical thermodynamics is accurately applied only to reversible processes, although bending of this rule still leads to acceptable results.
Carathéodory's work is somewhat inaccessible because of all the mathematics, but another Greek thermodynamicist developed a modification of the classical Boltzmann entropy that's easy to understand. Constantino Tsallis (b. 1943) is a physicist who's a naturalized citizen of Brazil, but he was born in Athens, Greece, lived in Argentina, and he was awarded his doctotal degree in physics from the University of Paris-Orsay.
Everyone who has taken a few physics or chemistry courses is familiar with Boltzmann's entropy formula,
in which S is the entropy, KB is the Boltzmann constant (1.38062 x 10-23 joule/kelvin), and Ω is the number of states accessible to the system. This equation is actually a simplification of the Boltzmann-Gibbs entropy when every state has the same probability pi; viz,
These equations define a maximum entropy, since the equal probability means that you can stuff an atom or molecule into any accessible state.
At this point, we might wonder what will happen when some states are correlated with each other, a condition for which the entropy will be less than the maximum. That's what Tsallis did in a 1988 paper in which he defined what's now known as the Tsallis entropy, Sq, given by the following equation:
in which the parameter, q, is called the entropic-index. This equation becomes the Boltzmann-Gibbs entropy equation when we take the limit as q approaches one. It's been a while since I did limits in Calculus, so I'll just take this on faith.
There are presently 69 papers posted on arXiv having "Tsallis entropy" in their titles. What's so important about Tsallis entropy? Boltzmann entropy applies only to systems in equilibrium, and the Boltzmann entropy is an extensive function; that is, it depends on how much matter we have in our system. In non-equilibrium systems we need another way to look at entropy, and now we have Tsallis entropy, which is non-extensive. As can be imagined, a lot of people have a problem with a non-extensive entropy.
Tsallis entropy has been shown to produce better results than Boltzmann entropy for the analysis of some systems in biology, nuclear physics, finance, music, and linguistics; and the correlations in DNA sequences. Despite these successes, there's been criticism that the entropic-index q acts merely as a fitting parameter. There's also an argument that Tsallis entropy violates the zeroth law of thermodynamics. That's the law that states that systems in thermal equilibrium with another system are in equilibrium with each other.
In the end, it appears that physicists just need to know whether to apply one entropy or another to a given system. It's not unlike knowing that you should use classical mechanics for bowling balls and quantum mechanics for atoms.
|Some founders of thermodynamics. Clockwise from the upper left, Carno, Boyle, Kelvin, Joule, Boltzmann, Gibbs, Maxwell, and Clausius. Images, via Wikimedia Commons, Carnot, Boyle, Kelvin, Joule, Clausius, Maxwell, Gibbs, Boltzmann. (Click for larger image).|
- Constantino Tsallis, "Possible generalization of Boltzmann-Gibbs statistics," Journal of Statistical Physics, vol. 52, nos. 1-2 (July, 1988), pp. 479-487.
- H. V. Ribeiro, E. K. Lenzi, R. S. Mendes, G. A. Mendes, and L. R. da Silva, "Symbolic Sequences and Tsallis Entropy," arXiv Preprint Server, January 16, 2014.
- Jon Cartwright, "Roll over, Boltzmann," Physics World, May, 2014, pp. 31-35.
Permanent Link to this article
Linked Keywords: Thermodynamics; scientist; nationality; Sadi Carno (1796-1832); Robert Boyle (1627-1691); William Thomson, Lord Kelvin (1824-1907); James Prescott Joule (1818-1889); Rudolf Clausius (1822-1888); James Clerk Maxwell (1831-1879); Josiah Willard Gibbs (1839-1903); Ludwig Boltzmann (1844-1906); Founders of Thermodynamics; Greece; Western intellectual tradition; philosophy; science; thermodynamics; thermodynamicist; Constantin Carathéodory (1873-1950); profession; mathematics; applied mathematics; axiom; axiomatic; classical mechanics; reversible process; irreversibility; uiform; military dress; youth; Royal Military Academy; Belgium; doctoral advisor; Hermann Minkowski; special relativity; Wikimedia Commons; Boltzmann's entropy formula; Boltzmann entropy; Constantino Tsallis (b. 1943); physicist; naturalization; naturalized citizen; Brazil; Athens; Argentina; doctorate; doctotal degree; physics; University of Paris-Sud; University of Paris-Orsay; chemistry; course; entropy; Boltzmann constant; joule; kelvin; Boltzmann-Gibbs entropy; probability; atom; molecule; correlation; correlated; Tsallis entropy; parameter; limit; Calculus; scientific literature; paper; arXiv; thermodynamic equilibrium; intensive and extensive properties; extensive function; matter; Centro Brasileiro de Pesquisas Físicas; biology; nuclear physics; finance; music; linguistics; nucleic acid sequence; DNA sequence; curve fit; fitting; zeroth law of thermodynamics; physicist; bowling ball; quantum mechanics.
Books by Dev Gualtieri
Blog Article Directory on a Single Page
- Tsallis Entropy - October 29, 2014
- Sagittarius A - October 27, 2014
- Liquid Nanoparticles - October 24, 2014
- Truffles and Chinese Gooseberries - October 22, 2014
- Coiling - October 20, 2014
- Perfect Solar Absorber - October 17, 2014
- Diamond Thread - October 15, 2014
- Trust in Science - October 13, 2014
- Nobel Prize for Blue LEDs - October 10, 2014
- Veiled Venus - October 8, 2014
- Martin Perl (1927 - 2014) - October 6, 2014
- World Population - October 3, 2014
- Salt Corrosion - October 1, 2014
- Temperature Cycle Energy-Harvesting - September 29, 2014
- Quilting Semiconductors - September 26, 2014
- The Pleiades - September 24, 2014
- A Graphene Drum - September 22, 2014
- The 2014 MacArthur Fellows - September 19, 2014
- Early Oxygen - September 17, 2014
- Ammonia Synthesis - September 15, 2014
- Splitting Water Inexpensively - September 12, 2014
- A Willing Suspension of Disbelief - September 10, 2014
- The Mass of the Milky Way - September 8, 2014
- Whiter Whites - September 5, 2014
- Magnetic Refrigeration - September 3, 2014
- Peak Copper - September 1, 2014
- Terracotta Army - August 29, 2014
- Molybdenum Disulfide for DNA Sequencing - August 27, 2014
- The Branly Coherer - August 25, 2014
- Cultural Mobility - August 22, 2014
- Stardust - August 20, 2014
- Nickel Hair - August 18, 2014
- Solar Steam - August 15, 2014
- Music Analysis - August 13, 2014
- Boron Buckyballs - August 11, 2014
- Gaffer Drone - August 8, 2014
- Silly Putty - August 6, 2014
- Lunar Caves - August 4, 2014
- Bowtie Nanoantennas - August 1, 2014
- The Miller Experiment - July 30, 2014
- Gradient Materials - July 28, 2014
- Heronian Triangles - July 25, 2014
- Thin Optical Doubler - July 23, 2014
- Geomagnetic Reversal - July 21, 2014
- Coupled Lasers - July 18, 2014
- Batteries from Sand - July 16, 2014
- Materials Supply Chain - July 14, 2014
- Vacuum Transistors - July 11, 2014
- Particle Clumping - July 9, 2014
- A Magnetic Hose - July 7, 2014
- The Jacobi Method Updated - July 4, 2014
- Strong Cellulose Filaments - July 2, 2014
- Chemical Evolution - June 30, 2014
- The Man in the Moon's Missing Twin - June 27, 2014
- Thermogalvanic Cell - June 25, 2014
- Kevlar® Inventor, Stephanie Kwolek - June 23, 2014
- Fairy Rings and Fairy Circles - June 20, 2014
- Engineering Ice Cream - June 18, 2014
- Photonic Thermometry - June 16, 2014
- One Man's Food... - June 13, 2014
- Graphene from Nickel - June 11, 2014
- P = NP - June 9, 2014
- Our Lumpy Universe - June 6, 2014
- Godzilla, King of the GMOs - June 4, 2014
- Mass-Energy Equivalence - June 2, 2014
- STEM Indicators - May 30, 2014
- Keywords - May 28, 2014
- Graphene Production - May 26, 2014
- Algal Phytochromes - May 23, 2014
- Simulating the Universe - May 21, 2014
- Fifty Years of BASIC - May 19, 2014
- You Are What You Write - May 16, 2014
- The Bugs of Summer - May 14, 2014
- Norbert Wiener - May 12, 2014
- Let's Do The Twist - May 9, 2014
- Droplet Lens - May 7, 2014
- Shotgun Pi - May 5, 2014
- A Younger Man's Game? - May 2, 2014
Deep Archive 2006-2008