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ScienceWeek
PHYSICS: SMALL-SYSTEM NONEQUILIBRIUM THERMODYNAMICS
The following points are made by C. Bustamante et al (Physics Today 2005 July):
1) Small systems found throughout physics, chemistry, and biology manifest striking properties as a result of their tiny dimensions. Examples of such systems include magnetic domains in ferromagnets, which are typically smaller than 300 nm; quantum dots and biological molecular machines that range in size from 2 to 100 nm; and solidlike clusters that are important in the relaxation of glassy systems and whose dimensions are a few nanometers. Researchers nowadays are interested in understanding the properties of such small systems. For example, they are beginning to investigate the dynamics of the biological motors responsible for converting chemical energy into useful work in the cell. Those motors operate away from equilibrium, dissipate energy continuously, and make transitions between steady states.
2) Until the early 1990s, researchers had lacked experimental methods to investigate various properties of small systems, such as how they exchange heat and work with their environments. The development of modern techniques of microscopic manipulation has changed the experimental situation. In parallel, during the past decade, theorists have developed several results collectively known as fluctuation theorems (FTs), some of which have been experimentally tested. The much-improved experimental access to the energy fluctuations of small systems and the formulation of the principles that govern both energy exchanges and their statistical excursions are starting to shed light on the unique properties of microscopic systems. Ultimately, the knowledge physicists are gaining with their new experimental and theoretical tools may serve as the basis for a theory of the nonequilibrium thermodynamics of small systems.
3) Thermodynamics describes energy exchange processes of macroscopic systems: Objects as varied as liquids, magnets, superconductors, and even black holes comply with its laws. In macroscopic systems, behavior is reproducible and fluctuations (deviations from the typically observed average behavior) are small. It is only under some special conditions that thermal fluctuations produce readily detectable consequences in macroscopic systems. Well-known examples include the opalescence of light in a fluid at its critical point and the blue color of the sky, which is a result of light scattering.
4) As a system's dimensions decrease, fluctuations away from equilibrium begin to dominate its behavior. In particular, in a nonequilibrium small system, thermal fluctuations can lead to observable and significant deviations from the system's average behavior. Therefore, such systems are not well described by classical thermodynamics. Systems of this type abound in the laboratory, where scientists are building motors with dimensions of less than 100 nm, and in the cell, where the biological function and efficiency of molecules such as the molecular motor kinesin are determined by molecular size.[1-5]
References (abridged):
1. For a review, see F. Ritort, Seminaire Poincare 2, 193 (2003)
2. D. J. Evans, D. J. Searles, Phys. Rev. E 50, 1645 (1994); G. Gallavotti, E. G. D. Cohen, Phys. Rev. Lett. 74, 2694 (1995). For a review of fluctuation theorems, see D. J. Evans, D. J. Searles, Adv. Phys. 51, 1529 (2002)
3. D. J. Evans, E. G. D. Cohen, G. P. Morriss, Phys. Rev. Lett. 71, 2401 (1993)
4. C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997)
5. G. E. Crooks, Phys. Rev. E 60, 2721 (1999)
Physics Today http://www.physicstoday.org
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THEORETICAL PHYSICS: ON NONEQUILIBRIUM PHYSICS
The following points are made by David Ruelle (Physics Today 2004 May):
1) Statistical mechanics attempts to explain the macroscopic properties of matter in terms of the interactions of its microscopic constituents. To be precise, one starts with a mathematical description of the time evolution of a system of many particles -- molecules for example. The evolution is deterministic, defined by a Hamiltonian H that may be classical or quantum.
2) Statistical mechanics was developed at the end of the 19th century by, among others, James Clerk Maxwell (1831-1879), Ludwig Boltzmann (1844-1906), and Josiah Willard Gibbs (1839-1903). Equilibrium statistical mechanics is concerned with certain states of matter that appear macroscopically at rest, in equilibrium, and that are microscopically a superposition of states (i), with probabilities p(subi).
3) If the Hamiltonian of the system describes particles that repel each other at short distances and have negligible interaction at large distances, then an asymptotic "thermodynamic limit" exists in which the volume and number of particles tend to infinity at fixed density and temperature. Or, instead of fixing the temperature, one can fix the energy density: The two types of description are equivalent. In favorable cases, one can study phase transitions, the critical points, and so forth. Difficult problems remain, though, such as proving the existence of crystal phases, but the successes of equilibrium statistical mechanics during the 20th century were stupendous.
4) After the foundations were laid in the 19th century, equilibrium statistical mechanics developed slowly at first, then rapidly. The 19th century also saw the development of basic ideas of nonequilibrium statistical mechanics. Boltzmann showed how entropy, which measures the amount of microscopic randomness compatible with a macroscopic description, could statistically only increase with time. The philosophically correct views of Boltzmann, however, did not lead to a simple, general, useful prescription that could be systematically used to attack nonequilibrium problems. Note that instead of nonequilibrium, people also speak of dissipation or irreversibility. The three terms refer to different aspects of the same thing: When a system is outside of equilibrium, it dissipates energy as heat, and that happens with an irreversible increase of the entropy. Mechanical, chemical, and electrical energy may all be dissipated in irreversible processes.
5) In the above description, the dynamics -- that is, time evolution -- is eliminated. All one needs to know are the energies of the various states. In nonequilibrium problems, the dynamics cannot be forgotten. Consider specific heat and heat resistance. For specific heat, one asks how much heat needs to be put in a lump of matter to increase its temperature by one degree. For heat resistance, one asks what temperature difference needs to be put across a lump of matter to force a unit of energy to pass through the lump per unit time. The two problems may not sound very different, but the first is equilibrium, does not involve dynamics, and is relatively easy. The second is nonequilibrium, involves dynamics, and is difficult.(1-5)
References (abridged):
1. D. Ruelle, in Mathematics: Frontiers and Perspectives, V. Arnold et al., eds., American Mathematical Society, Providence, RI (2000), p. 251
2. H. Spohn, J. L. Lebowitz, Commun. Math. Phys. 54, 97 (1977)
3. See, for example, A. Haro, R. de la Llave, Phys. Rev. Lett. 85, 1859 (2000)
4. For a brief discussion of the independent introduction by William Hoover and by Dennis Evans of the isokinetic thermostat, see D. J. Evans, G. P. Morriss, Statistical Mechanics of Nonequilibrium Liquids, Academic Press, London (1990), section 5.2
5. G. Gallavotti, E. G. D. Cohen, Phys. Rev. Lett. 74, 2694 (1995); J. Stat. Phys. 80, 931 (1995)
Physics Today http://www.physicstoday.org
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STATISTICAL PHYSICS: EQUILIBRIUM STATISTICAL MECHANICS APPLIED TO NONEQUILIBRIUM SYSTEMS
Notes by ScienceWeek:
Statistical mechanics (statistical physics) is a quantitative approach to the average behavior of a system containing many particles, the approach derived from first principles and certain simplifying assumptions concerning the nature and interactions of the particles in the system. It is the most successful approach to the behavior of physical systems containing many particles, but its application has been limited to systems at or near thermodynamic equilibrium.
The following points are made by David A. Egolf (Science 2000 287:101):
1) The author points out that statistical mechanics describes the macroscopic physical properties of matter through a probabilistic rather than a detailed knowledge of microscopic dynamics, and that the theory has been applied successfully to a wide variety of equilibrium systems, ranging from simple molecular gases to *white dwarf stars. Statistical mechanics has provided a theoretical understanding of the phases of matter, the transitions between phases, and the deep property of universality that unifies the descriptions of continuous transitions in systems physically quite distinct (e.g., magnets and gases). In nature, however, many systems are not in equilibrium, including, for example, large-scale flows in the atmosphere, the evolution of ecological systems, and the transport of energy in biological cells. None of these situations can presently be understood with equilibrium statistical mechanics.
2) Although the theory of equilibrium statistical mechanics has been developed to extend it to systems only slightly perturbed away from equilibrium (for which a quantitative description of the evolution of the system is well-approximated with only linear terms), in deterministic systems driven far from equilibrium (where nonlinearities are important), theoretical progress has been limited to relatively simple situations. In particular, theorists have not yet developed an understanding of the intriguing phenomenon of spatially extended *chaos, which is typically characterized by disordered arrays of defects, patches of uncorrelated regions, and a chaotic dynamics that persists indefinitely. This remarkable behavior has been found in large, deterministic, far-from-equilibrium systems as varied as convecting horizontal fluid layers, chemical reaction-diffusion systems, colonies of microorganisms, and *fibrillating heart tissue. These disparate systems often display strikingly similar macroscopic features and behaviors, which suggests the question of whether one can construct a statistical predictive theory of phases and transitions applicable to such chaotic far-from-equilibrium systems.
3) The author reports that in his own computer-analysis study, at intermediate coarse-grained scales, of a simple far-from-equilibrium spatially extended chaotic model system, a number of equilibrium properties, including *ergodicity and *detailed balance, were found to be recovered by the system, which indicates, the author suggests, that the macroscopic behavior of some far-from-equilibrium systems might be understood in terms of equilibrium statistical mechanics.
4) The essential idea resulting from this work and proposed by the author is that simple far-from-equilibrium *dissipative and extensively chaotic systems "can recover certain equilibrium properties at coarse-grained scales with the underlying chaotic dynamics serving as a temperature bath." The author concludes: "The system studied here possesses some important differences from true equilibrium systems. Perhaps the most intriguing is that the effective noise strength (or temperature) is internally generated and dependent on the state of the system, rather than imposed by an external temperature bath. This difference poses a challenge for explorations of the *second law of thermodynamics in these systems."
Science http://www.sciencemag.org
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Notes by ScienceWeek:
white dwarf stars: White dwarf stars are extremely dense and compact stars that have undergone gravitational collapse. Such stars, the final stage in the evolution of low-mass stars after they have lost their outer layers, are approximately the size of Earth, but with a mass approximately that of the Sun.
chaos: In this context, the term "chaos" refers to unpredictable behavior arising in a system that obeys deterministic laws but exhibits unpredictability. The essential idea is that in certain systems small perturbations may produce a cascade of larger perturbations, so that eventually the behavior of such systems cannot be predicted from prior states no matter if the systems appear simple and obey deterministic laws.
fibrillating heart tissue: Heart muscle fibrillation, which is a dysfunction, is an extremely rapid desynchronized contraction or twitching of individual muscle fibers in a muscle.
ergodicity: In general, ergodicity is a property of dynamic systems containing a random variable (stochastic systems): a system is said to be ergodic if it tends in probability to a limiting form which is independent of the initial conditions.
detailed balance: The principle of detailed balancing (also called the principle of microscopic reversibility) states that in equilibrium the probability (frequency) of the transition of any microscopic part of a system from state A to state B equals the probability (frequency) of the transition from state B to state A.
dissipative: In general, a dissipative system is a system that loses energy by conversion of energy into heat.
second law of thermodynamics: This law concerns the direction that a natural process can take, and the law can be stated in various ways, for example: a) heat cannot be transferred from one body to a second body at a higher temperature without producing some other effect; b) the entropy of a closed system increases with time.
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