Boltzmann Equation

Type of physical science: Chaos theory, Thermodynamics, Classical physics

Field of study: Thermodynamics

The Boltzmann equation is the fundamental defining relationship quantifying the concept of entropy, showing how entropy is related to microscopic disorder and chaos. The equation shows entropy to be a statistical variable that links macroscopic observations of the second law of thermodynamics with microscopic molecular behavior and probabilities.

Overview

The Boltzmann equation is the fundamental relationship that links microscopic disorder and chaos with the macroscopic concept of entropy, the variable used to quantify the second law of thermodynamics. It was Ludwig Boltzmann, an ardent defender of atomic theory at a time when it was out of favor, who realized that entropy could be expressed quantitatively in terms of the natural logarithm of the total number of microstates available to a system in equilibrium with a constant of proportionality now known as the Boltzmann constant, 1.381 × 10²³; joules per Kelvin.

The microstate is a microscopic concept tied to atomic theory. It is a description of the particular arrangement of the internal structure of a system in terms of the positions and velocities of all the particles that make up that system. The total number of possible equilibrium microstates available to a system corresponds to a particular macrostate, the macroscopically measured state of a system described in terms of thermodynamic quantities such as temperature, pressure, and volume. These macroscopic quantities are related to the average microscopic properties of a system—average kinetic energy, average number density, and positions of the atoms and molecules making up the system. There are usually many possible microstates that yield the same average equilibrium values for temperature, pressure, and volume. As an analogy, there are many different combinations of integers that can yield a given average value. Hence, there are many microstates corresponding to a macrostate.

A thermodynamic universe is composed of a system and an environment. No energy can flow into or out of a thermodynamic universe, but energy can flow between a system and its environment. It is necessary to consider the entropy of a system and of its environment.

One of Boltzmann's strokes of genius was to envision a way to quantify the entropy of a system and its environment in an additive way. The number of microstates corresponding to a macrostate describing a system and its environment are related in a multiplicative fashion. For each microstate of the system, there can be many microstates of the environment yielding the same equilibrium macrostate, and vice versa. Hence, the total number of microstates of a system-plus-environment corresponding to an equilibrium macrostate is equal to the multiplicative product of the possible number of microstates of the system times the possible number of microstates of the environment. Since the logarithm of a product is equal to the sum of the logarithms of the individual factors, Boltzmann chose to express the concept of entropy in terms of the logarithms of the number of microstates of the system and environment corresponding to their equilibrium macrostate. In this way, the total entropy of a system and its environment is equated to the entropy of the system plus the entropy of its environment.

The second law of thermodynamics can then be expressed qualitatively as "the total entropy of a thermodynamic universe cannot decrease with time." According to this statement, a system will spontaneously evolve to the macrostate of maximum entropy. Hence, a drop of ink will diffuse throughout a volume of water, but will not spontaneously reverse this process.

The second law of thermodynamics is often misused as applying only to a system, leading to the false conclusion that the entropy of a system cannot decrease with time. In fact, the entropy of a system can decrease, but only at the expense of an even greater increase in entropy of the environment, so that the sum does not decrease with time. Simple examples of this can be observed in the operation of such devices as refrigerators, air conditioners, and heat pumps.

The Boltzmann equation is not only central to the study of thermodynamics; it is also the essential link between the predictions of classical and quantum-statistical mechanics—the study of systems in terms of atomic and molecular behavior—and the observations of macroscopic thermodynamics. Classical thermodynamics is the one truly empirical science, with absolutely no assumptions as to internal structure of the material making up systems. On the other hand, statistical mechanics attempts to reproduce the predictions of thermodynamics based upon microscopic models of structure and composition. Since thermodynamics is based solely upon macroscopic observation, any assumptions or models of the nature of the microscopic composition of a system must yield predicted behavior that agrees with the predictions of thermodynamics.

Classical statistical mechanics and quantum statistical mechanics are the study of systems in terms of their postulated atomic and molecular constituents. Boltzmann was the "founder" and principal developer of statistical mechanics, a field which is also referred to as "kinetic theory." The Boltzmann equation ties the predicted behavior of these models to the observed second law of thermodynamics.

An example (highly simplified) can be used to illustrate the significance of the Boltzmann equation. Consider a box with two distinguishable particles free to move around in it (call them the "red" and "green" particles). A particle may be located in the left half or the right half of the box. Each possible configuration of these particles (for example, red in the left half and green in the right half of the box) is a microstate.

Disregarding velocities and assuming only two positions, in this case there are four possible microstates available to the particles in this system: the red and green particles in the left-hand side of the box; the red and green particles in the right-hand side of the box; red in the left and green in the right; and green in the left and red in the right.

On the other hand, the distribution of one particle in each half of the box is a macrostate. There are two ways in which this macrostate can be achieved: red in the left and green in the right, or green in the left and red in the right. For the macroscopic observer, both cases of one particle on the left and one particle on the right will yield the same equilibrium thermodynamic macrostate. The thermodynamic properties of this box are not sensitive as to which is the red particle and which is the green particle.

Hence, there are two microstates corresponding to this macrostate. For two particles, this does not seem to be very dramatic. However, if there were a hundred particles in the box, there would be approximately one million million million million million different ways in which they could be distributed between the left and right sides. Of all the possible different microstates available to the system, approximately one-tenth of them correspond to the macrostate with fifty particles on the left and fifty particles on the right. This number of microstates greatly exceeds the number of microstates corresponding to any other macrostate arrangement of particles. The number of microstates corresponding to a hundred particles on the left side and zero particles on the right side is only one. It is far more likely that the particles will be distributed evenly, with fifty on the left and fifty on the right, than that they will be distributed a hundred on the left and zero on the right.

As the number of particles in a system increases, the measurable fluctuations (for example, forty-nine particles on the left and fifty-one particles on the right) from the equilibrium macrostate become less and less probable. It may be possible—even likely—that for ten particles, three will be found on the left side and seven on the right side at some given instant. However, the probability of finding thirty particles on the left and seventy on the right is nearly infinitesimal.

Boltzmann's equation relates the number of microstates to the entropy of the system. The second law of thermodynamics states that a system and environment will spontaneously evolve to the highest entropy level. This corresponds to the system and environment evolving to the macrostate corresponding to the largest number of microstates. It is really a matter of probabilities and statistics. It is far more likely that there will be fifty particles in each half of the box (or very small random fluctuations around fifty in each half) than it is that the distribution will evolve to a hundred particles preferentially moving to the left half of the box.

As the number of particles in a system increases, the probability of significant fluctuations from the macrostate of largest microstates becomes less and less. It is possible that two particles can be found in the left half of the box. However, in the case of a room filled with one million million million million air molecules, it is far, far more likely that the air molecules will be approximately evenly distributed throughout the room than that all of the molecules will be spontaneously located in the left half of the room. The molecules have many more microstates yielding the macrostate of molecules evenly distributed throughout the room. Even a 1 percent fluctuation from this distribution has only a tiny probability of occurring. The Boltzmann equation's relation of entropy to the microstates of a system thus demonstrated that the second law of thermodynamics is really a principle of the evolution of a system (or more generally, a thermodynamics universe) based upon probability and statistics.

Ludwig Boltzmann never used the well-known formulation of what is referred to as the "Boltzmann equation"; however, it is his concept relating entropy to the microstates of a system that is formulated in the equation. Indeed, the Boltzmann equation is inscribed on his tombstone in the central cemetery of Vienna. Although simple in construction, it is arguably one of the most remarkable formulas in nature.

Applications

Engineering applications are more apt to use the classical formulation of entropy than the Boltzmann equation. However, for areas of research and development incorporating the atomic level—and even information theory—the Boltzmann equation has important applications.

It is with the development and application of classical statistical mechanics, quantum statistical mechanics, and chaos theory that the Boltzmann equation finds its principal uses. On the basis of statistics, the Boltzmann equation shows that processes spontaneously progress or transform in certain directions but not other directions. For example, a drop of ink placed in a bowl of water will spontaneously diffuse throughout the water, that is, to a macrostate with the highest number of microstates. However, the diffused ink will not spontaneously collect and jump out of the water as a drop of ink—a macrostate with far fewer microstates. As ridiculous as this example may sound, it does not violate any of the other laws and conservation principles of physics. It is only the second law of thermodynamics that addresses this issue, and the Boltzmann equation shows that it is really a result of statistics and probability.

Another macroscopic example is the case of breaking a rack of fifteen billiard balls with a cue ball. Striking a rack of fifteen tightly arranged balls in a triangle with a speeding cue ball will tend to scatter all the balls around the table. Yet sixteen chaotically moving billiard balls never come to rest in a final position with fifteen balls in a tight triangle and the cue ball at the other end of the table.

Perhaps the most important application of the Boltzmann equation is that it is the principle link that shows how atomic theory can account for and predict macroscopically observed thermodynamic behavior of systems. It is the Boltzmann equation that supplied the theoretical "ammunition" that eventually completely overthrew continuum theory, that is, the theory that matter is infinitely divisible into smaller and smaller parts. The concept of microstates, macrostates, and the employment of these concepts to quantify entropy are central to an atomic theory, a theory that sees matter as finitely divisible to its smallest unit, the atom. It is the Boltzmann equation that supplies atomic theory with its strongest experimental and theoretical support.

Context

The Boltzmann equation is one of several vital contributions that Ludgwig Boltzmann made to statistical mechanics and to physics in general. Maxwell-Boltzmann statistics, distribution of energy states, the Boltzmann constant, the H-theorem, and the Boltzmann transport equation are but a sampling of his genius and of his important contributions to science and technology.

It is ironic that in his lifetime Boltzmann's ideas were not well received. Atomic theory was very much out of favor. Indeed, his suicide in 1906 is thought to have been the result of despondency brought on by the ridicule of his atomic ideas by the scientific community of the time. Only a year after his death, however, the experimental evidence became so overwhelming that the reigning beliefs of continuum theory were completely overturned. Boltzmann's ideas regarding atomic theory and the statistical mechanics accompanying it were vindicated and became the standard model of the scientific community.

The Boltzmann equation does not appear in popular physics literature with the frequency or volume given to other, perhaps more glittering, areas of physics. However, its significance ranks with that of any of the other great principles of physics. Somewhat like the director of a film, the Boltzmann equation describes and directs that important link between what the observer sees and what is thought to be happening behind the scenes at the microscopic level. It may very well be the most important link between atomic theory and the macroscopic world.

Students of physics, however, generally do not encounter the Boltzmann equation until after several years of studying "introductory" classical physics, atomic physics, and often a variety of other areas. Consequently, the accessibility of Boltzmann's concept to the general audience is somewhat limited, and its exposure to the general public is not nearly as broad as that of other great ideas of physics.

Principal terms

Boltzmann constant: a fundamental constant of nature critical to the study of thermodynamics and statistical mechanics

disorder: a description of entropy at the microscopic level; disorder is a quantitative measure of the lack of order (or chaos) of a system, environment, or thermodynamic universe

entropy: a statistical variable linking the observed direction of processes with probable molecular behavior

first law of thermodynamics: essentially, the principle of conservation of energy as applied to a closed universe: Any energy that flows into (or out of) a system must flow out of (or into) the environment

macrostate: the equilibrium state of a thermodynamic system, based upon the average molecular properties of the system

microscopic: in this context, the level of atomic dimensions, that is, in the range of one billionth of a meter or less

microstate: the specific microscopic state of a thermodynamic system, given by specific molecular properties of each individual molecule in the system; many different microstates can correspond to the same macrostate, since many different microstates can yield the same average molecular properties

second law of thermodynamics: a law of nature associated with the observation that many transformation processes occur in only one direction: For example, an ice cube at room temperature will spontaneously melt and come into thermoequilibrium with the room temperature; however, water will not spontaneously shed energy and form an ice cube in the same room

thermodynamic environment: that region outside a thermodynamic system that makes up the rest of the thermodynamic universe

thermodynamic system: an entity of interest characterized by three thermodynamic variables, one of which is always temperature; the other variables may be pressure and volume but are not restricted to these

thermodynamic universe: a system plus environment that is completely isolated from anything else; no matter or energy can flow into or out of a thermodynamic universe

Bibliography

Atkins, P. W. The Second Law. New York: W. H. Freeman, 1984. An excellent book on the second law of thermodynamics. Discusses the interrelations among the Boltzmann equation, entropy, and chaos.

Cromer, Alan. Physics for the Life Sciences. Hightstown, N.J.: McGraw-Hill, 1977. Cromer's treatment of the statistical formulation of the second law of thermodynamics is illuminating. The discussion, which should be quite understandable to the general reader, shows very clearly the relationship of the Boltzmann equation to the second law of thermodynamics. Chapter 12, in particular, is strongly recommended.

Davies, Paul. About Time: Einstein's Unfinished Revolution. New York: Simon & Schuster, 1995. Although the entries regarding Boltzmann and his work in entropy and statistical mechanics are brief, the discussion relating Boltzmann's work to the development of physics is significant.

Pais, Abraham. Subtle Is the Lord . . . : The Science and the Life of Albert Einstein. New York: Oxford University Press, 1982. Pages 60-75 give a surprisingly good treatment of Boltzmann and his work. Not only is the Boltzmann equation discussed and placed in a historical perspective, but its influence on the work of Albert Einstein is also presented.

Prigogine, Ilya, and Isabelle Stengers. Order Out of Chaos. New York: Bantam Books, 1984. This highly readable work incorporates Boltzmann's equation as well as his work in general into the development of thermodynamics and statistical mechanics.

Segre, Emilio. From X-Rays to Quarks: Modern Physicists and Their Discoveries. New York: W. H. Freeman, 1980. Of the books listed in this bibliography, this book is probably the best treatment of the development of physics for the general reader with some mathematics background. Includes a discussion of the significance of the Boltzmann equation in that development.