Mathematics of energy

Summary: Mathematics is used to study energy and energy conservation as well as to develop new sources of energy.

The concept of energy and transportation of energy are central to the survival of any civilization. As mathematical physicist Ludwig Boltzmann noted, “Available energy is the main object at stake in the struggle for existence and the evolution of the world.” At the start of the twenty-first century, human beings have accessed or created many forms of energy and power production, including coal-fired and oil-fired power plants, solar heating plants, wind farms, nuclear power plants, geothermal sources of heat, hydroelectric power produced by dams, biofuels that store solar energy, and tidal energy produced by gravitational interactions between Earth and the moon.

There are also potentially disruptive energy sources, including natural events, such as lightning, volcanoes, and earthquakes. Some global sources of energy and power that remain to be tapped by humans include the atmosphere’s expansion and contraction, ocean currents, and sea level differences. Various calculations of energy, including chemical reactions and nuclear reactions, invoke the principle of conservation of energy. In relativistic or quantum terms, the conservation of mass-energy is also important. Energy, work, and quantity of heat are all expressed in “joules,” a measure of work named for physicist James Joule. There is a vast array of energy problems that mathematicians research, and mathematics makes many contributions to energy issues.

Energy, Defined

Energy is found in nearly every system or process in the universe: mechanics, chemicals, heat, electricity, nuclear processes, and quantum effects. Mathematician and scientist René Descartes studied mechanics; centuries later, mathematician and philosopher Gottfried Wilhelm Leibniz criticized his ideas and developed what are referred to today as “kinetic energy,” “potential energy,” and “momentum.” In mechanics, the kinetic energy (E) of an object is expressed as

where m is the object’s mass and v is its velocity. Another form of energy found in mechanics is the energy of position called “potential energy.” It has the units of joules. An example is the potential energy defined as work done in the compression of a coiled spring. The sum of all the kinetic and potential energies within a system comprises the mechanical energy of the system. Energy may be a conserved quantity within a closed system, or it may change forms, such as mechanical energy being converted to heat by friction. How energy in a system is measured is important. As noted, mechanical energy is measured as the sum of kinetic energy and potential energy, or energies of motion and position. Chemical energy is measured by the heat energy released in chemical reactions. Electrical energy is measured by work done in a system.

Energy Conservation

In general, the amount of energy of various types can be equated to an equivalent amount of heat energy. On an experimental scale, heat energy is the ability of work done to raise the temperature of water. The joule is a measure of thermodynamic energy and is the common unit of energy. James Prescott Joule is credited with experiments in the mid-1800s that demonstrated that work done on a system can be converted into heat. His experiments and those of others eventually led to the realization and statement of the “principle of conservation of energy” as a hypothesis, which was proved in certain restricted settings and generalized by induction. In 1865, mathematical physicist Rudolf Clausius worked on thermodynamics and stated his first law as, “The energy of the universe is constant.” The principle of conservation of energy applies not only to certain mechanical systems but is also seen widely in systems where other forms of energy are considered. Thus, heat energy is produced by combustion and friction, radiant energy is from light and other forms of radiation, and chemical energy is stored in fuels and electrical energy. The principle is continually tested in new situations. This testing led to discoveries in the twentieth century in atomic physics. In the International System of Units, Le Système International d’Unités (SI), a joule is defined as a newton-meter, named for Isaac Newton. The systematic study of the relation of various physical quantities through an analysis of their dimensions is the subject of dimensional analysis. Richard Feynman noted, “For those who want some proof that physicists are human, the proof is in the idiocy of all the different units which they use for measuring energy.”

One energy issue that has been important to mathematicians, philosophers, and physicists is the relationship between matter and energy. Some physicists wanted to assign matter-like properties to energy, such as Wilhelm Wien, who considered that energy might have a traceable motion. Mathematician William Clifford thought of matter and energy as types of curvatures. In the theory of special relativity of 1905, Albert Einstein proved an equivalence of mass and energy as expressed in his famous equation E=mc², where E is the energy equivalent of mass m, and c denotes the speed of light, 299,792,458 meters per second. There is no process available to human beings at the start of the twenty-first century in which matter can be converted completely into radiant energy.

For example, in a nuclear explosion, only a tiny fraction of nuclear material is converted into energy. The only known process of annihilating matter is to pair a particle of matter with a particle of anti-matter, with the result that two photons are formed with energies that are equivalent to the energies of the particles. This process is on a quantum scale. Fusion is one process for partially converting mass into energy and occurs naturally in stars. Many controlled fusion experiments have been performed but in the process of producing fusion, a greater amount of input energy is needed for the reaction than is ultimately released by the reaction. Only in uncontrolled thermonuclear explosions are large amounts of energy released by fusion.

Fusion

Scientists continue to explore novel sources of energy and power from sources that entail motion, heat, quantum uncertainty and other natural physical phenomena. One possible source of power is controlled fusion reactions, hot or cold. Controlled hot fusion reactions have not yet reached a break-even point where the energy of the reaction exceeds the energy input needed to trigger the reaction.

There are ongoing fusion experiments that use various solids and liquids with energy pumped into them by lasers in which fusion occurs but the fusion is not self-sustaining. The main problem is the energy input and inherent danger in heating suitable substances to temperatures at which fusion between atoms of hydrogen isotopes can occur. The hydrogen is in the form of deuterium or tritium, and the temperatures reached through compression must be on the order of millions of degrees, and there are often energetic byproducts that are dangerous to objects and people. In contrast to hot fusion, cold fusion (also known as “low-energy nuclear reactions” among the twenty-first-century research community) is the fusion of atoms at close to room temperature, generally through the use of supersaturated metal hydrides. These reactions produce heat, helium, and a very low level of neutrons. The energy output is greater than the input, leading many scientists and others to investigate this process as a viable solution to the energy needs of the future. Chemists Martin Fleischmann and Stanley Pons were the first, in 1989, to publicly announce that they had achieved cold fusion. Many competing scientific and mathematical models have been developed to explain how cold fusion works but many researchers and others remain skeptical regarding its existence or viability.

Other Mathematical Applications

Mathematicians and other scientists have long studied the various aspects of energy. The concept of energy is fundamental to many scientific and business theories, applications, and disciplines. For instance, mathematicians have modeled energy trading in financial markets, which is quantitatively interesting because, in such applications, energy possesses unique attributes as a non-storable and non-fungible commodity. They have also worked to design efficient shutdown schedules for electronic systems to address concerns related to energy conservation. Mathematics is important for explaining the cosmic phenomenon of dark energy. This type of energy, often modeled as a scalar field and inferred in large part from observation and mathematical analysis of gravitational fields, has implications for theories and measurement of universe expansion and dark matter. On the other hand, mathematicians such as Blake Temple have used mathematics to attempt to disprove the existence of dark energy and posit alternative explanations. Others have investigated the geometry of symplectic energy. Mathematicians are also influential in energy research and policy making via work at federal agencies like the U.S. Department of Energy. Mathematician J. Ernest Wilkins was a fellow at the Department of Energy’s Argonne National Laboratory and physicist and mathematician Hermann Bondi was the chief scientific adviser to the Department of Energy. Mathematical analysis and computational methods have also been used to study energy problems related to equilibrium, stability, and energy transport.

Bibliography

Coopersmith, Jennifer. Energy, the Subtle Concept: The Discovery of Feynman’s Blocks from Leibniz to Einstein. New York: Oxford University Press, 2010.

Gerritsen, Margot. “Mathematics Awareness Month April 2009 Theme Essay: Mathematics in Energy Production.” http://www.mathaware.org/mam/09/essays/Margot‗EnergyMaths.pdf.

Greengard, Claude, and Andrzej Ruszczynski. Decision Making Under Uncertainty: Energy and Power. New York: Springer, 2010.

Society for Industrial and Applied Mathematics. “Fuel Cells, Energy Conversion, and Mathematics.” http://www.siam.org/about/news-siam.php?id=1605.

Veigele, William. How to Save Energy and Money at Home and on the Highway: The Mathematics and Physics of Energy Conservation and Reduction of Consumer Energy Costs. Boca Raton, FL: Universal Publishers, 2009.