Theorem In Statistical Mechanics Research Articles

Energy Transients in Harmonic Oscillator Systems

The approach to energy equilibration is discussed for a system of N isolated classical oscillators, coupled or uncoupled. Exact solutions can be given for the kinetic and potential energies vs time, for a simple boundary condition which distributes the potential energy equally between all the normal modes at zero time and which gives zero kinetic energy to all the normal-mode oscillators at zero time. Most solutions are ergodic but a special class is nonergodic, i.e., the potential energy vs time shows exact recursions. The half-life for approaching the equilibrium value of potential or kinetic energy is of the order of the reciprocal maximum frequency of the oscillators. The fluctuations from equilibrium are treated by an extension of the method of Rayleigh's problem of random walks. The results are of interest in connection with the ergodic theorem of statistical mechanics.

Physical Limitations on Ray Oscillation Suppressors

The question of whether it is possible to suppress ray oscillations in light waveguides is important for the design of light communications systems. With the help of Liouville's theorem of statistical mechanics it is shown that it is impossible to reduce simultaneously the amplitudes and the angles of ray oscillations if the ray originates in and returns to a region of low index of refraction. A reduction of both ray amplitudes and angles can be achieved only if the ray moves from a region of low to one of high index of refraction. Liouville's theorem is used to derive a condition relating the output position and slope of a ray which traverses an optical transformer to its input position and slope. With p <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> , x <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> denoting the canonically conjugate variables of the output ray and p <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> , x <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> those of the input ray, the condition derived from Liouville's theorem states that the Jacobian of the transformation is one.

Definition of Normal Product and Wick's Theorem in Statistical Mechanics

Definition of Normal Product and Wick's Theorem in Statistical Mechanics

Une démonstration simplifiée du théorème de wick en mécanique statistique

Wick's theorem in statistical mechanics is proved directly by using the commutation rules.

Energy Equation of the Atmosphere and Irreversible Reciprocity Theorem in StatisticalMechanics

Energy Equation of the Atmosphere and Irreversible Reciprocity Theorem in StatisticalMechanics

A Theorem in Statistical Mechanics

STATISTICAL mechanics considers any particular body at given temperature T as a member of a canonical ensemble, that is, the probability of finding it with energy E is given by where ω is the multiplicity of the level E of this body. We can take it that, for the body under test, P is a maximum with regard to any parameter n so that The use of this relation appears to simplify appreciably the treatment of some problems in statistical mechanics. I have not, however, been able to find it mentioned anywhere in the known treatises on the subject.

A tentative statistical theory of Macleod’s equation for surface tension, and the parachor

Macleod’s (1923) empirical formula for the surface tension σ of a liquid in equilibrium with its own vapour, namely σ = const. ( ρ liq . ─ ρ vap ) 4 , (1) where the ρ ’s are densities in g./c. c., must impress everyone with the feeling that such a simple and successful relation must have some equally simple foundation in thermodynamic or statistical theory. This feeling is only intensified when one realizes the wide use to which this formula has been put by Sugden and his followers. Sugden (1930) writes M *σ ¼ / ρ liq . ─ ρ vap . = P , (2) where M * is the (chemical) molecular weight of the substance, and calls P the parachor . The comparative study of the parachors for various liquids seems to have proved fruitful. This study is described by Sugden as the comparative study of molecular volumes at equal surface tensions, and therefore, to the best of our opportunities, at equal “internal pressures”. We need not here enquire into the precise significance (if any) of the phrase ‘‘internal pressure”. It is sufficient to recognize that the quantity P defined by (2) is in fact a constant (very nearly) for any given substance, independent of the temperature over a wide range, from the critical temperature T c downwards—and having recognized this important empirical fact, to attempt to derive it as a theorem in statistical mechanics applied to a reasonable model, and to give a formula for P in terms of molecular diameters and (or) intermolecular forces.

Adsorption Isotherms. Critical Conditions

1. In a recent note I showed that Langmuir's adsorption isothermwhere θ is the fraction of the surface covered by adsorbed gas, p the gas pressure in equilibrium with it, and A (T) a specified function of the temperature, can be derived as a theorem in statistical mechanics without any appeal to the mechanism of deposition and re-evaporation. Necessary and sufficient assumptions for the truth of (1) are that the atoms (or molecules) of the gas are adsorbed as wholes on to definite points of attachment on the surface of the adsorber, that each point of attachment can accommodate one and only one adsorbed atom, and that the energies of the states of any adsorbed atom are independent of the presence or absence of other adsorbed atoms on neighbouring points of attachment. Under these assumptions the explicit form of (1) iswhere m is the mass of the adsorbed atom or molecule, bg(T) the partition function for its internal states in the gas phase, and vs(T) the partition function for its set of adsorbed states. These sets of states are to be so specified that the energy zero is assigned tot the lowest state of each set in constructing bg(T) and vs(T), and then X is the energy required to transfer a molecule from the lowest adsorbed state tot the lowest gas state. Quite another adsorption isotherm was shown to hold when adsorption of a molecule takes place as atoms and requires two or more points of attachment.