Statistical mechanics

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Microscopic particle motion

Statistical mechanics arose out of the development of classical thermodynamics. It is a mathematical framework applying methods of statistics and the theory probability to large assemblies of microscopic particles. It explains the macroscopic behavior of such ensembles. The founding of the field is generally credited to James Clerk Maxwell, Ludwig Boltzmann and Josiah Willard Gibbs. While classical thermodynamics is primarily concerned with thermodynamic equilibrium, statistical mechanics has been applied in non-equilibrium statistical mechanics to the issues of microscopically modeling the speed of irreversible processes driven by imbalances, such as chemical reactions.


  • To know the quantum mechanical state of a system implies, in general, only statistical restrictions on the results of measurements. It seems interesting to ask if this statistical element be thought of as arising, as in classical statistical mechanics, because the states in question are averages over better defined states for which individually the results would be quite determined. These hypothetical 'dispersion free' states would be specified not only by the quantum mechanical state vector but also by additional 'hidden variables' - 'hidden' because if states with prescribed values of these variables could actually be prepared, quantum mechanics would be observably inadequate.
    • John Stewart Bell, "On the problem of hidden variables in quantum mechanics" Reviews of Modern Physics (1966)
  • In contemplating the papers Einstein wrote in 1905, I often find myself wondering which of them is the most beautiful. It is a little like asking which of Beethoven’s symphonies is the most beautiful. My favorite, after years of studying them, is Einstein’s paper on the blackbody radiation. [...] Part of being a great scientist is to know—have an instinct for—the questions not to ask. Einstein did not try to derive the Wien law. He simply accepted it as an empirical fact and asked what it meant. By a virtuoso bit of reasoning involving statistical mechanics (of which he was a master, having independently invented the subject over a three-year period beginning in 1902), he was able to show that the statistical mechanics of the radiation in the cavity was mathematically the same as that of a dilute gas of particles. As far as Einstein was concerned, this meant that this radiation was a dilute gas of particles—light quanta. But, and this was also characteristic, he took the argument a step further. He realized that if the energetic light quanta were to bombard, say, a metal surface, they would give up their energies in lump sums and thereby liberate electrons from the surface in a predictable way, something that is called the photoelectric effect. [...] In the first place, not many physicists were even interested in the subject of blackbody radiation for at least another decade. Kuhn has done a study that shows that until 1914 less than twenty authors a year published papers on the subject; in most years there were less than ten. Planck, who was interested, decided that Einstein’s paper was simply wrong.
  • Newton and his theories were a step ahead of the technologies that would define his age. Thermodynamics, the grand theoretical vision of the nineteenth century, operated in the other direction with practice leading theory. The sweeping concepts of energy, heat, work and entropy, which thermodynamics (and its later form, statistical mechanics) would embrace, began first on the shop floor. Originally the domain of engineers, thermodynamics emerged from their engagement with machines. Only later did this study of heat and its transformation rise to the heights of abstract physics and, finally, to a new cosmological vision.
    • Adam Frank, About Time: Cosmology and Culture at the Twilight of the Big Bang (2011)
  • The Schrödinger equation, which is at the heart of quantum theory, is applicable in principle to both microscopic and macroscopic regimes. Thus, it would seem that we already have in hand a non-classical theory of macroscopic dynamics, if only we can apply the Schrödinger equation to the macroscopic realm. However, this possibility has been largely ignored in the literature because the current statistical interpretation of quantum mechanics presumes the classicality of the observed macroscopic world to start with. But the Schrödinger equation does not support this presumption. The state of superposition never collapses under Schrödinger evolution.
  • In the consistent-histories approach, the classical limit can be studies by using appropriate subspaces of the quantum Hilbert space as a "coarse graining," analogous to dividing up phase space into nonoverlapping cells in classical statistical mechanics. This coarse graining can then be used to construct quantum histories. It is necessary to show that the resulting family of histories is consistent, so that the probabilities assigned by quantum dynamics make good quantum mechanical sense. Finally, one needs to show that the resulting quantum dynamics is well approximated by appropriate classical equations.
  • [I]n the nineteenth century, even the theory of heat could be reduced to mechanics by the assumption that heat really consists of a complicated statistical motion of the smallest parts of matter. By combining the concepts of the mathematical theory of probability with the concepts of Newtonian mechanics Clausius, Gibbs and Boltzmann were able to show that the fundamental laws in the theory of heat could be interpreted as statistical laws following from Newton's mechanics when applied to very complicated mechanical systems.
  • In the history of Science it is possible to find many cases in which the tendency of Mathematics to express itself in the most abstract forms has proved to be of ultimate service in the physical order of ideas. Perhaps the most striking example is to be found in the development of abstract Dynamics. The greatest treatise which the world has seen, on this subject, is Lagrange's Mécanique Analytique, published in 1788. ...conceived in the purely abstract Mathematical spirit ...Lagrange's idea of reducing the investigation of the motion of a dynamical system to a form dependent upon a single function of the generalized coordinates of the system was further developed by Hamilton and Jacobi into forms in which the equations of motion of a system represent the conditions for a stationary value of an integral of a single function. The extension by Routh and Helmholtz to the case in which "ignored co-ordinates" are taken into account, was a long step in the direction of the desirable unification which would be obtained if the notion of potential energy were removed by means of its interpretation as dependent upon the kinetic energy of concealed motions included in the dynamical system. The whole scheme of abstract Dynamics thus developed upon the basis of Lagrange's work has been of immense value in theoretical Physics, and particularly in statistical Mechanics... But the most striking use of Lagrange's conception of generalized co-ordinates was made by Clerk Maxwell, who in this order of ideas, and inspired on the physical side by... Faraday, conceived and developed his dynamical theory of the Electromagnetic field, and obtained his celebrated equations. The form of Maxwell's equations enabled him to perceive that oscillations could be propagated in the electromagnetic field with the velocity of light, and suggested to him the Electromagnetic theory of light. Heinrich Herz, under the direct inspiration of Maxwell's ideas, demonstrated the possibility of setting up electromagnetic waves differing from those of light only in respect of their enormously greater length. We thus see that Lagrange's work... was an essential link in a chain of investigation of which one result... gladdens the heart of the practical man, viz. wireless telegraphy.
  • In the history of sciences, important advances often come from... the recognition that two hitherto separate observations can be viewed from a new angle and seen to represent nothing but different facets of one phenomenon. Thus, terrestrial and celestial mechanisms became a single science with Newton's laws. Thermodynamics and mechanics were unified through statistical mechanics, as were optics and electromagnetism through Maxwell's theory of magnetic field, or chemistry and atomic physics through quantum mechanics. Similarly different combinations of the same atoms, obeying the same laws, were shown by biochemists to compose both the inanimate and animate worlds. ...
    Despite such generalizations, however, large gaps remain... Following the line from physics to sociology, one goes from simpler to the more complex objects... from the poorer to the richer empirical content, as well as from the harder to the softer system of hypotheses and experimentation. ...Because of the hierarchy of objects, the problem is always to explain the more complex in terms and concepts applying to the simpler. This is the old problem of reduction, emergence, whole and parts... an understanding of the simple is necessary to understand the more complex, but whether it is sufficient is questionable. ...the appearance of life and later of thought and language—led to phenomena that previously did not exist... To describe and to interpret these phenomena new concepts, meaningless at the previous level, are required. ...At the limit total reductionism results in absurdity. ...explaining democracy in terms of the structure and properties of elementary particles... is clearly nonsense.
    • François Jacob, "Evolution and Tinkering," Science (June 10, 1977) Vol. 196, No. 4295
  • The kinetic theory of gases is a small branch of physics which has passed from the stage of excitement and novelty into staid maturity. ...Formerly it was hoped that the subject of gases would ultimately merge into a general kinetic theory of matter; but the theory of condensed phases... today, involves an elaborate and technical use of wave mechanics, and for this reason it is best treated as a subject in itself.
    The scope of the present book is, therefore, the traditional kinetic theory of gases. ...[A]n account has been included of the wave-mechanical theory, and especially of the degenerate Fermi-Dirac case... There is also a concise chapter on statistical mechanics, which... may be of use as an introduction... [T]he discussion of electrical phenomena has been abbreviated... the latter voluminous subject is best treated separately. ...[F]undamental parts have been explained... [as] to be within the reach of college juniors and seniors. The... wave mechanics and statistical mechanics... are of graduate grade. ...[A] number of carefully worded theorems have been inserted in the guise of problems, without proof... to give... a chance to apply... lines of attack exemplified in the text.
    To facilitate use as a reference book, definitions have been repeated freely, I hope not ad nauseam. ...Ideas have been drawn freely from ...books such as ...of Jeans and Loeb...
    • Earle Hesse Kennard, Kinetic Theory of Gases With an Introduction to Statistical Mechanics (1938) Preface.
  • The need for a fundamentally different approach to the study of physical processes at the molecular level motivated the development of relevant statistical methods, which turned out to be applicable not only to the study of molecular processes (statistical mechanics), but to a host of other areas such as the actuarial profession, design of large telephone exchanges, and the like. In statistical methods, specific manifestations of microscopic entities (molecules, individual telephone sites, etc.) are replaced with their statistical averages, which are connected with appropriate macroscopic variables. The role played in Newtonian mechanics by the calculus, which involves no uncertainty, is replaced in statistical mechanics by probability theory, a theory whose very purpose is to capture uncertainty of a certain type.
    • George Klir, B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications (1995), p. 1-2.
  • The path integral is a formulation of quantum mechanics equivalent to the standard formulations, offering a new way of looking at the subject which is, arguably, more intuitive than the usual approaches. Applications of path integrals are as vast as those of quantum mechanics itself, including the quantum mechanics of a single particle, statistical mechanics, condensed matter physics and quantum field theory. ...
    It is in quantum field theory, both relativistic and nonrelativistic, that path integrals (functional integrals is a more accurate term) play a much more important role, for several reasons. They provide a relatively easy road to quantization and to expressions for Green’s functions, which are closely related to amplitudes for physical processes such as scattering and decays of particles. The path integral treatment of gauge field theories (non-abelian ones, in particular) is very elegant: gauge fixing and ghosts appear quite effortlessly. Also, there are a whole host of nonperturbative phenomena such as solitons and instantons that are most easily viewed via path integrals. Furthermore, the close relation between statistical mechanics and quantum mechanics, or statistical field theory and quantum field theory, is plainly visible via path integrals.
  • There is an interesting analogy... with the philosophy of the natural sciences, which has flourished under the combined influence of both general methodology and classical metaphysical questions (realism vs. antirealism, space, time, causation, etc.) interacting with detailed case studies in... (physics, biology, chemistry, etc.)... [C]ase studies both historical (studies of Einstein's relativity, Maxwell's electromagnetic theory, statistical mechanics, etc.). By contrast, with few exceptions, philosophy of mathematics has developed without the corresponding detailed case studies.
    • Paolo Mancosu, The Philosophy of Mathematical Practice (2008) Introduction, p. 2.
  • Another crucial point is that MOND as we know it now is arguably only an approximate 'effective field theory' that approximates some more fundamental scheme at a deeper stratum — some 'FUNDAMOND' — conceptually, in a similar way to thermodynamics being an approximation of the statistical-mechanics, microscopic description.
  • With the growing importance of models in statistical mechanics and in field theory, the path integral method of Feynman was soon recognized to offer frequently a more general procedure of enforcing the first quantization instead of the Schrödinger equation. To what extent the two methods are actually equivalent, has not always been understood... [T]here are few nontrivial models which permit deeper insight into their connection. However, the exactly solvable cases... the Coulomb potential and the harmonic oscillator... point the way: For scattering problems the path integral seems particularly convenient, whereas for the calculation of discrete eigenvalues the Schrödinger equation [is preferable]. ...[P]otentials with degenerate vacua ...arise recently studied models of large spins.
  • You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, no one really knows what entropy really is, so in a debate you will always have the advantage.
  • Carnot's Principle. ...If physical phenomena were due exclusively to the movements of atoms whose mutual attraction depended only on the distance, it seems that all these phenomena should be reversible; if all the initial velocities were reversed, these atoms, always subjected to the same forces, ought to go over their trajectories in the contrary sense, just as the earth would describe in the retrograde sense this same elliptic orbit which it describes in the direct sense, if the initial conditions of its motion had been reversed. On this account, if a physical phenomenon is possible, the inverse phenomenon should be equally so, and one should be able to reascend the course of time. Now, it is not so in nature, and this is precisely what the principle of Carnot teaches us; heat can pass from the warm body to the cold body; it is impossible afterward to make it take the inverse route and to reestablish differences of temperature which have been effaced. Motion can be wholly dissipated and transformed into heat by friction; the contrary transformation can never be made except partially.
    We have striven to reconcile this apparent contradiction. If the world tends toward uniformity, this is not because its ultimate parts, at first unlike, tend to become less and less different; it is because, shifting at random, they end by blending. For an eye which should distinguish all the elements, the variety would remain always as great; each grain of this dust preserves its originality and does not model itself on its neighbors; but as the blend becomes more and more intimate, our gross senses perceive only the uniformity. This is why for example, temperatures tend to a level, without the possibility of going backwards.
    A drop of wine falls into a glass of water; whatever may be the law of the internal motion of the liquid, we shall soon see it colored of a uniform rosy tint, and however much from this moment one may shake it afterwards, the wine and the water do not seem capable of again separating. Here we have the type of the irreversible physical phenomenon : to hide a grain of barley in a heap of wheat, this is easy; afterwards to find it again and get it out, this is practically impossible. All this Maxwell and Boltzmann have explained; but the one who has seen it most clearly, in a book too little read because it is a little difficult to read, is Gibbs, in his 'Elementary Principles of Statistical Mechanics.’
  • The only important variables of interest must involve averaging over many of the degrees of freedom. Statistical mechanics is the formalization of this intuitive concept. The problems to be addressed... are threefold: under what circumstances can the properties of a physical system be defined by the behavior of an appropriate small set of variables, what are the appropriate sets of relevant variables, and how can one calculate the properties of the system in terms of these variables.
    • Ivo Sachs, Siddhartha Sen, James Sexton Elements of Statistical Mechanics With an Introduction to Quantum Field Theory and Numerical Simulation (2006) Ch. 1 The Problem, p. 2.
  • I thought of calling it 'information,' but... Von Neumann told me, 'You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, no one really knows what entropy really is, so in a debate you will always have the advantage.'
  • [G]ame theory has already established itself as an essential tool in the behavioral sciences, where it is widely regarded as a unifying language for investigating human behavior. Game theory's prominence in evolutionary biology builds a natural bridge between the life sciences and the behavioral sciences. And connections have been established between game theory and the two most prominent pillars of physics: statistical mechanics and quantum theory. ...[M]any physicists, neuroscientists, and social scientists... are... pursuing the dream of a quantitative science of human behavior. Game theory is showing signs of... an increasing important role in that endeavor. It's a story of exploration along the shoreline separating the continent of knowledge from an ocean of ignorance... a story worth telling.
    • Tom Siegfried, A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature (2006) Preface
  • Maxwell, and then Boltzmann, and then... J. Willard Gibbs consequently expended enormous intellectual effort in devising... statistical mechanics, or... statistical physics. The uses... extend far beyond gases... describing electric and magnetic interactions, chemical reactions, phase transitions... and all other manner of exchanges of matter and energy.
    The success... has driven the belief among many physicists that it could be applied with similar success to society. ...[E]verything from the flow of funds in the stock market to the flow of traffic on interstate highways ...
    • Tom Siegfried, A Beautiful Math (2006) Ch. 7: Quetelet's Statistics and Maxwell's Molecules, pp. 142-143.
  • Thermodynamics is more like a mode of reasoning than a body of physical law. ...we can think of thermodynamics as a certain pattern of arrows that occurs again and again in very different physical contexts, but, wherever this pattern of explanation occurs, the arrows can be traced back by the methods of statistical mechanics to deeper laws and ultimately to the principles of elementary particle physics. ...the fact that a scientific theory finds applications to a wide variety of different phenomena does not imply anything about the autonomy of this theory from deeper physical laws.
  • As Oliver Cromwell said to the General Assembly of the Church of Scotland, "I beseech you, in the bowels of Christ, think it possible that you might be mistaken." Life and the affairs of the living are so tangled, the world not only stranger than we imagine but stranger than we can imagine, that all questions are conundrums, no answers "correct." Is it certain that parallel lines never meet? No. Does water freeze at 32 degrees Fahrenheit? Only probably. Shall I marry? Who can say? And yet the world's work must be done. One Oblomov is enough. Thus we learn a conventional certitude, acting as though all were light by blinking the shadow. A simple proof demonstrates that parallel lines do meet, but, on the assumption that they do not, the architect builds the skyscraper. Despite his knowledge of statistical mechanics, the engineer designs the refrigerator to maintain a constant temperature of 31 degrees. Le cœur a ses raisons que la raison ne connait pas [the heart has its reasons that reason does not know], and families are raised.

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