Statistical mechanics Boltzmann

In statistical mechanics, Boltzmann defined entropy of a system as... 

The equations of quantum statistical mechanics for a system of non-identical particles, for which all solutions of the wave equations are accepted, are closely analogous to the equations of classical statistical mechanics (Boltzmann statistics). The quantum statistics resulting from the acceptance of only antisymmetric wave functions is considerably different. This statistics, called Fermi-Dirac statistics, applies to many problems, such as the Pauli-Sommerfeld treatment of metallic electrons and the Thomas-Fermi treatment of many-electron atoms. The statistics corresponding to the acceptance of only the completely symmetric wave functions is called the Bose-Einstein statistics. These statistics will be briefly discussed in Section 49. 

It may be necessary to amend the underlying theory before the reduction process. In order to derive thermodynamics from classical statistical mechanics, Boltzmann was, for example, obliged to introduce the quasiergodic hypothesis. There are vague speculations that quantum theory may have to be amended for systems with a very large number of particles [17]. 

The concept of entropy is present in many disciplines. In statistical mechanics, Boltzmann introduced entropy as a measure of the number of microscopic ways that a given macroscopic state can be realized. A principle of nature is that it prefers systems that have maximum entropy. Shannon has also introduced entropy into communications theory, where entropy serves as a measure of information. The role of entropy in these fields is not disputed in the scientific community. The validity, however, of the Bayesian approach to probability theory and the principle of maximum entropy in this, remains controversial. 

It is of interest in the present context (and is useful later) to outline the statistical mechanical basis for calculating the energy and entropy that are associated with rotation [66]. According to the Boltzmann principle, the time average energy of a molecule is given by... 

Another important accomplislnnent of the free electron model concerns tire heat capacity of a metal. At low temperatures, the heat capacity of a metal goes linearly with the temperature and vanishes at absolute zero. This behaviour is in contrast with classical statistical mechanics. According to classical theories, the equipartition theory predicts that a free particle should have a heat capacity of where is the Boltzmann constant. An ideal gas has a heat capacity consistent with tliis value. The electrical conductivity of a metal suggests that the conduction electrons behave like free particles and might also have a heat capacity of 3/fg,... 

Nearly ten years ago, Tsallis proposed a possible generalization of Gibbs-Boltzmann statistical mechanics. [1] He built his intriguing theory on a reexpression of the Gibbs-Shannon entropy S = —k Jp r) np r)dr written... 

The average of the step function, using the action for a Boltzmann weight can be pursued by standard statistical mechanics. It may require more elaborate sampling techniques such as the Umbrella sampling [20]. 

The Boltzmann distribution is fundamental to statistical mechanics. The Boltzmann distribution is derived by maximising the entropy of the system (in accordance with the second law of thermodynamics) subject to the constraints on the system. Let us consider a system containing N particles (atoms or molecules) such that the energy levels of the... 

The work on gas theory had many extensions. In 1865 Johann Josef Loschmidt used estimates of the mean free path to make the first generally accepted estimate of atomic diameters. In later papers Maxwell, Ludwig Boltzmann, and Josiah Willard Gibbs extended the rrratherrratics beyorrd gas theory to a new gerreralized science of statistical mechanics. Whenjoined to quantum mechanics, this became the foundation of much of modern theoretical con-derrsed matter physics. 

Second Derivation of the Boltzmann Equation.—The derivation of the Boltzmann equation given in the first sections of this chapter suffers from the obvious defect that it is in no way connected with the fundamental law of statistical mechanics, i.e., LiouviUe s equation. As discussed in Section 12.6of The Mathematics of Physics and Chemistry, 2nd Ed.,22 the behavior of all systems of particles should be compatible with this equation, and, thus, one should be able to derive the Boltzmann equation from it. This has been avoided in the previous derivation by implicitly making statistical assumptions about the behavior of colliding particles that the number of collisions between particles of velocities v1 and v2 is taken proportional to /(v.i)/(v2) implies that there has been no previous relation between the particles (statistical independence) before collision. As noted previously, in a... 

J. G. Kirkwood and J. Boss, The Statistical Mechanical Basis of the Boltzmann Equation, in I. Frigogine, ed., Transport Processes in Statistical Mechanics, pp. 1-7, Interscience Publishers, Inc., New York, 1958. Also, J. G. Kirkwood, The Statistical Mechanical Theory of Transport Processes I. General Theory, J, Chem, Phys, 14, 180 (1946) II. Transport in Gases, J, Chem. Phys, 15, 72 (1947). 

In his book States of Matter, [(1985), Prentice Hall, Dover] David L. Goodstein writes Ludwig Boltzmann, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics. Perhaps it will be wise to approach the subject cautiously. ... 

Various statistical treatments of reaction kinetics provide a physical picture for the underlying molecular basis for Arrhenius temperature dependence. One of the most common approaches is Eyring transition state theory, which postulates a thermal equilibrium between reactants and the transition state. Applying statistical mechanical methods to this equilibrium and to the inherent rate of activated molecules transiting the barrier leads to the Eyring equation (Eq. 10.3), where k is the Boltzmann constant, h is the Planck s constant, and AG is the relative free energy of the transition state [note Eq. (10.3) ignores a transmission factor, which is normally 1, in the preexponential term]. 

This is a law about the equilibrium state, when macroscopic change has ceased it is the state, according to the law, of maximum entropy. It is not really a law about nonequilibrium per se, not in any quantitative sense, although the law does introduce the notion of a nonequilibrium state constrained with respect to structure. By implication, entropy is perfectly well defined in such a nonequilibrium macrostate (otherwise, how could it increase ), and this constrained entropy is less than the equilibrium entropy. Entropy itself is left undefined by the Second Law, and it was only later that Boltzmann provided the physical interpretation of entropy as the number of molecular configurations in a macrostate. This gave birth to his probability distribution and hence to equilibrium statistical mechanics. 

The aim of this section is to give the steady-state probability distribution in phase space. This then provides a basis for nonequilibrium statistical mechanics, just as the Boltzmann distribution is the basis for equilibrium statistical mechanics. The connection with the preceding theory for nonequilibrium thermodynamics will also be given. 

The work of Ludwig Boltzmann (1844-1906) in Vienna led to a better understanding, and to an extension, of the concept of entropy. On the basis of statistical mechanics, which he developed, the term entropy experienced an atomic interpretation. Boltzmann was able to show the connections between thermodynamics and the phenomenon of order and chance events he used the term entropy as a measure... 

In equilibrium statistical mechanics involving quantum effects, we need to know the density matrix in order to calculate averages of the quantities of interest. This density matrix is the quantum analog of the classical Boltzmann factor. It can be obtained by solving a differential equation very similar to the time-dependent Schrodinger equation... 

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