Statistical theory of heat

The earliest hint that physics and information might be more than just casually related actually dates back at least as far as 1871 and the publication of James Clerk Maxwell s Theory of Heat, in which Maxwell introduced what has become known as the paradox of Maxwell s Demon. Maxwell postulated the existence of a hypothetical demon that positions himself by a hole separating two vessels, say A and B. While the vessels start out being at the same temperature, the demon selectively opens the hole only to either pass faster molecules from A to B or to pass slower molecules from B to A. Since this results in a systematic increase in B s temperature and a lowering of A s, it appears as though Maxwell s demon s actions violate the second law of thermodynamics the total entropy of any physical system can only increase, or, for totally reversible processes, remain the same it can never decrease. Maxwell was thus the first to recognize a connection between the thermodynamical properties of a gas (temperature, entropy, etc.) and the statistical properties of its constituent molecules. 

W. P. Hettinger jr., N. R. Larson, and J. A. Wethington jr. Specific heat of synthetic high polymers. I. A study of polyethylene including a statistical theory of crystallite length. J. Chem. Phys. 20, 781—790 (1952). 

The theory of heat has not been reduced to statistical mechanics "How can the zeroth law of thermodynamics be derived from statistical mechanics "... 

Maxwell, J.H., Theory of Heat, Longman Green, London, 1871. Brush, S.G., Statistical Physics and the Atomic Theory of Matter, Princeton University Press, Princeton, NJ, 1983. van der Waals, J.H. PhD Thesis, University of Leiden, 1873. 

Taylor GA. (1935) Statistical theory of turbulence. Proc. R. Soc. Lond. A, 151 421 444. Whitaker S. (1972) Forced convection heat transfer correlations for flow in pipes, past flat plates, single cylinders, single spheres and for flow in packed beds and mbe bundles. AICHEJ, 18 361-370. 

Clausius, Rudolf (1850). On the Motive Power of Heat, and on the Laws which can be deduced from it for the Theory of Heat. Poggendorff s Annalen der Physik. Crawford, F.H. (1963). Heat, Thermodynamics and Statistical Physics. Rupert Hart-Davis, London, Harcourt, Brace World, Inc. 

It is clear that one needs to know the heat capacities of a substance as a function of temperature and pressure in order to calculate the entropy and other thermodynamic quantities. A detailed understanding of the theory of heat capacities (which requires statistical mechanics) is beyond the scope of this book. Here we shall only give a brief outline of Peter Debye s theory for the heat capacities of solids, an approach that leads to an approximate general theory. The situation is more complex in liquids because there is neither complete molecular disorder, as in a gas, nor is there long-range order, as in a solid. 

Only statistical methods can be of use when studying the thermodynamic and structural characteristics of solution, such as heat of dissolution, energy of solvation, energy of solvent reorganization and radial distribution functions (probabilities of solvent molecules being located at a given distance from the solute molecule). In the statistical theory of solution, the energy of interaction with the medium is written as [11] ... 

Various forms of models are used in research, either as a starting point or modified/ developed as part of the research itself. Types of models include (1) a theoretical model, based on a theory which may exist prior to undertaking the research (ie, the theory of heat transfer) or be developed or modified during the research (eg, mechanical properties of stitches in a weft knit textile Webster et al., 1998) (2) a physical model (eg, layers of fabric in a series of textile layers simulating infant bedding assembly) (Wilson et al., 1999) or (3) a statistical model (eg., to elucidate dominant factors and/or relationships among variables in the experiment). 

Figure 5 The derived kinetic energy release from the energy-selected Ar2CO + ions as a function of the trimer ion internal energy. The solid line is a calculated kinetic energy release based on the statistical theory of dissociation phase space theory (PST) or the version of PST due to C.E. Klots. AP is the threshold energy for ArCO+ formation. The onset leads to a heat of formation of the trimer ion. Reproduced with permission from Mahnert J, Baumgartel H and Weltzel KM (1997) The formation of ArCO+ ions by dissociative Ionization of argon/carbon monoxide clusters. Journal of Physical Chemistry 07 6667-6676.

Another example of structural insight comes from the heat capacities of tri-atomic molecules. According to statistical theory the heat capacities (at constant volume) of linear and bent molecules are 6.5A b and G.Ofcs, respectively. Thus support for the linear structure of CO2 and the bent structure of SO2 comes both from heat capacity measurements and statistical thermodynamics. 

By the end of the nineteenth century the development of the wave theory of light, commenced by Young and Fresnel, seemed to have reached its culmination in the brilliant theoretical work of Maxwell. In another branch of physics the theory of heat leading on to the kinetic theory of gases and to statistical mechanics as developed by Clausius, Boltzmann, Maxwell, and Gibbs also seemed to be complete in most of its essential points. This encouraged Michelson to write in 1899 ... 

In statistical mechanics (e.g. the theory of specific heats of gases) a degree of freedom means an independent mode of absorbing energy by movement of atoms. Thus a mon-... 

Spectroscopic Methods, [Biological] Applications of Spectroscopy, EPR, Recent Advances in (Smaller). Spectroscopy, Infrared, Use in Biology (Lecomte). Spectroscopy of Transition-Group Complexes (Jorgensen) Statistical-Mechanical Theory of Transport Processes. X. The Heat of Transport in Binary Liquid Systems (Bearman, Kirkwood, Fixman). ... 

Experiments indicate that the smooth variations of thermodynamic properties (e.g., V, Ky, and the specific heat at constant pressure Cp) with temperature are intermpted by the kinetic process of glass formation, leading to cooling rate dependent kinks in these properties as a function of temperature. In our view, these kinks cannot be described by an equilibrium statistical mechanical theory, but rather are a challenge for a nonequilibrium theory of glass formation. Nonetheless, some insight into the origin of these kinks and the qualitative... 

Statistical-Mechanical Theory of Transport Processes. X. The Heat of Transport in Binary Liquid Systems (Bearman, Kirkwood, ... 

It is noteworthy that Gibbs himself was acutely aware of the qualitative failures of 19th-century molecular theory (as revealed, for example, by erroneous classical predictions of heat capacities Sidebar 3.8). In the preface to his Elementary Principles in Statistical Mechanics, Developed with Especial Reference to the Rational Foundation of Thermodynamics (published in the last year of his life), Gibbs wrote ... 

Hendrik Antoon Lorentz, from Leyden (Holland), presided the conference, whose general theme was the Theory of Radiation and the Quanta. The conference5 was opened with speeches by Lorentz and Jeans, one on Applications of the Energy Equipartition Theorem to Radiation, the other on the Kinetic Theory of Specific Heat according to Maxwell and Boltzmann. In their talks, the authors explored the possibility of reconciling radiation theory with the principles of statistical mechanics within the classical frame. Lord Rayleigh, in a letter read to the... 

We see that each of our three sciences of heat has its own advantage s. A properly trained physicist or chemist should know all three, to be able to use whichever is most suitable in a given situation. We start witli thermodynamics, since it is the most general and fundamental method, taking up thermodynamic calculations in the next chapter. Following that we treat statistical mechanics, and still later kinetic theory. Only then shall we be prepared to make a real study of the nature of matter. 

We notice that at low temperatures the specific heat of a system with continuous energy levels, obeying the Fermi statistics, is proportional to the temperature. We shall later see that this formula has applications in the theory of metals. 

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