http://www.compadre.org/STP/ The latest material additions to the Statistical and Thermal Physics. en-US Copyright 2012, ComPADRE.org stp@compadre.org stp@compadre.org Wed, 31 Aug 2011 11:33:33 EST http://blogs.law.harvard.edu/tech/rss http://www.compadre.org/portal/services/images/LogoSmallSTP.gif http://www.compadre.org/STP/ 125 35 http://www.compadre.org/STP/items/detail.cfm?ID=11442 The author develops analogies between economic systems and thermodynamics, and shows how economic quantities can characterize the state of an economic system in equilibrium. We argue that just as a physical system in thermodynamic equilibrium requires a nonmechanical variable (the temperature) to specify its state, so does an economic system. In addition, both systems must have a corresponding conjugate quantity, the entropy. We also develop economic analogies to the free energy, Maxwell relations, and the Gibbs–Duhem relation. Assuming that economic utility can be measured, we develop an operational definition of an economic temperature scale. We also develop an analogy to statistical mechanics, which leads to Gaussian fluctuations. Thermo & Stat Mech/General http://www.compadre.org/STP/bulletinboard/Thread.cfm?ID=11442 Wed, 31 Aug 2011 11:33:33 EST http://www.compadre.org/STP/items/detail.cfm?ID=11442 http://www.compadre.org/STP/items/detail.cfm?ID=11441 The Legendre transform is a powerful tool in theoretical physics and plays an important role in classical mechanics, statistical mechanics, and thermodynamics. In typical undergraduate and graduate courses the motivation and elegance of the method are often missing, unlike the treatments frequently enjoyed by Fourier transforms. We review and modify the presentation of Legendre transforms in a way that explicates the formal mathematics, resulting in manifestly symmetric equations, thereby clarifying the structure of the transform. We then discuss examples to motivate the transform as a way of choosing independent variables that are more easily controlled. We demonstrate how the Legendre transform arises naturally from statistical mechanics and show how the use of dimensionless thermodynamic potentials leads to more natural and symmetric relations. Thermo & Stat Mech/General http://www.compadre.org/STP/bulletinboard/Thread.cfm?ID=11441 Wed, 31 Aug 2011 11:20:41 EST http://www.compadre.org/STP/items/detail.cfm?ID=11441 http://www.compadre.org/STP/items/detail.cfm?ID=11440 An analogy between black hole physics and thermodynamics is discussed, and equations similar to the usual partial differential relations of thermodynamics are found for black holes. The results can be used to supplement an undergraduate course on thermodynamics. Thermo & Stat Mech/General http://www.compadre.org/STP/bulletinboard/Thread.cfm?ID=11440 Wed, 31 Aug 2011 10:46:59 EST http://www.compadre.org/STP/items/detail.cfm?ID=11440 http://www.compadre.org/STP/items/detail.cfm?ID=11438 A large class of problems in statistical physics becomes much simpler when treated by means of the chemical potential. In this paper the chemical potential is applied to eight problems of physical interest, with particular emphasis on solid-state physics. The method can profitably be emphasized in upper division courses on statistical physics. Thermo & Stat Mech/Ensembles/Chemical Potential http://www.compadre.org/STP/bulletinboard/Thread.cfm?ID=11438 Wed, 31 Aug 2011 10:08:52 EST http://www.compadre.org/STP/items/detail.cfm?ID=11438 http://www.compadre.org/STP/items/detail.cfm?ID=11437 The author discusses the chemical potential, a topic that students invariably find difficult. Three "meanings" for the chemical potential are stated and then supported by analytical development. Two substantial applications, depression of the melting point and batteries, illustrate the chemical potential in action. The origin of the term "chemical potential" has its surprises, and a sketch of the history is given. Thermo & Stat Mech/Ensembles/Chemical Potential http://www.compadre.org/STP/bulletinboard/Thread.cfm?ID=11437 Wed, 31 Aug 2011 09:56:08 EST http://www.compadre.org/STP/items/detail.cfm?ID=11437 http://www.compadre.org/STP/items/detail.cfm?ID=11436 A simple experiment for the demonstration of percolation problems is reported. Measurements were performed on a sheet of conducting paper after randomly cutting out small square ‘sites’ at a concentration 1?p. The conductance and the area fraction covered by the infinite cluster were evaluated as a function of p. This experiment is appropriate as an introduction to the teaching of phase transitions. Thermo & Stat Mech/Phase Transitions/Critical Point http://www.compadre.org/STP/bulletinboard/Thread.cfm?ID=11436 Tue, 30 Aug 2011 17:24:12 EST http://www.compadre.org/STP/items/detail.cfm?ID=11436 http://www.compadre.org/STP/items/detail.cfm?ID=11435 The physics of stars is a goldmine of problems in statistical mechanics and thermodynamics. We discuss many examples that illustrate the possibility of deepening student’s knowledge of statistical mechanics by an introductory study of stars. The matter constituting the various stellar objects provides examples of equations of state for classical or quantum and relativistic or non-relativistic gases. Maximum entropy can be used to characterize thermodynamic and gravitational equilibrium which determines the structure of stars and predicts their instability above a certain mass. Contraction accompanying radiation induces either heating or cooling, which explains the formation of stars above a minimum mass. The characteristics of the emitted light are understood from blackbody radiation and from the Boltzmann–Lorentz kinetic equation for photons. The luminosity is governed by the transport of heat by photons from the center to the surface. Heat production by thermonuclear fusion is determined by microscopic balance equations. The stability of the steady state of stars is controlled by the interplay of thermodynamics and gravitation. Thermo & Stat Mech/Thermal Properties of Matter http://www.compadre.org/STP/bulletinboard/Thread.cfm?ID=11435 Tue, 30 Aug 2011 17:02:28 EST http://www.compadre.org/STP/items/detail.cfm?ID=11435 http://www.compadre.org/STP/items/detail.cfm?ID=11434 Thermodynamic states and processes involving light are discussed in which the chemical potential of light is nonzero. Light with nonzero chemical potential is produced in photochemical reactions, for example, in a light emitting diode. The chemical potential of black-body radiation becomes negative upon a Joule expansion. The isothermal diffusion of light is driven by the gradient in the chemical potential. These and other examples support the idea that light can be interpreted as a gas of photons, with properties similar to a material gas. Thermo & Stat Mech/Ensembles/Chemical Potential http://www.compadre.org/STP/bulletinboard/Thread.cfm?ID=11434 Tue, 30 Aug 2011 16:36:46 EST http://www.compadre.org/STP/items/detail.cfm?ID=11434 http://www.compadre.org/STP/items/detail.cfm?ID=11432 It is shown that the usual derivation of the heat capacity of an ideal free-electron gas is made invalid by the use of a divergent series. A plausible argument is offered to indicate that these series are asymptotic expansions which provide good approximations through their first few terms. The usual procedure is modified by the replacement of an infinite series by its first two terms plus a remainder whose bounds are estimated as being negligible to the accuracy required; a rigorous derivation of the well-known temperature dependence of the electronic heat capacity of an ideal Fermi gas results. Thermo & Stat Mech/Models/Fermi Gas http://www.compadre.org/STP/bulletinboard/Thread.cfm?ID=11432 Tue, 30 Aug 2011 16:20:52 EST http://www.compadre.org/STP/items/detail.cfm?ID=11432 http://www.compadre.org/STP/items/detail.cfm?ID=11431 The author reviews models of economic opinions, urban segregation, and language change and shows that the two-dimensional Ising model gives about the same results in each case. Thermo & Stat Mech/Models/Ising Model http://www.compadre.org/STP/bulletinboard/Thread.cfm?ID=11431 Tue, 30 Aug 2011 15:56:16 EST http://www.compadre.org/STP/items/detail.cfm?ID=11431 http://www.compadre.org/STP/items/detail.cfm?ID=11388 The authors compute the multiplicities (density of states) numerically for an Einstein solid - a collection of identical quantum harmonic oscillators. It is shown that if two such systems can exchange energy, some macrostates are overwhelmingly more probable than others. Graphs of the entropy versus the energy for the two systems are used to motivate the thermodynamic definition of temperature. Other simple properties of the Einstein solid and a system of noninteracting spins are also explored. Thermo & Stat Mech/Models/Einstein Model http://www.compadre.org/STP/bulletinboard/Thread.cfm?ID=11388 Thu, 11 Aug 2011 16:59:00 EST http://www.compadre.org/STP/items/detail.cfm?ID=11388 http://www.compadre.org/STP/items/detail.cfm?ID=11293 The radial distribution function is a measure of the spatial distribution of a system of particles. The authors discuss an experiment suitable for undergraduates that illustrates the meaning of the radial distribution function for a two-dimensional system of hard disks comprised of varying area fractions. Larger area fractions lead to an increase in the correlation length and the magnitude of the underlying particle–particle correlations. Thermo & Stat Mech/Statistical Physics http://www.compadre.org/STP/bulletinboard/Thread.cfm?ID=11293 Sat, 02 Jul 2011 13:25:19 EST http://www.compadre.org/STP/items/detail.cfm?ID=11293 http://www.compadre.org/STP/items/detail.cfm?ID=11292 The van der Waals equation of state together with Maxwell’s equal area rule, leads to a transcendental equation for the densities of the two coexisting phases. Gibbs solved that this equation can be solved in parametric form. The author shows that the parameter can be chosen to be the difference between the entropy per molecule in the vapor and the entropy per molecule in the liquid Δs. The parametric solution gives the thermodynamic properties of the two coexisting phases as functions of Δs. Thermo & Stat Mech/Phase Transitions/First-Order Transitions http://www.compadre.org/STP/bulletinboard/Thread.cfm?ID=11292 Sat, 02 Jul 2011 12:10:34 EST http://www.compadre.org/STP/items/detail.cfm?ID=11292 http://www.compadre.org/STP/items/detail.cfm?ID=11287 The renormalization group theory of continuous phase transitions is described in a way suitable for an undergraduate statistical physics course. Thermo & Stat Mech/Phase Transitions/Continuous Transitions http://www.compadre.org/STP/bulletinboard/Thread.cfm?ID=11287 Thu, 30 Jun 2011 10:01:41 EST http://www.compadre.org/STP/items/detail.cfm?ID=11287 http://www.compadre.org/STP/items/detail.cfm?ID=11286 It is shown that pressure melting cannot be responsible for the low friction of ice. Thermo & Stat Mech/Thermal Properties of Matter/Pressure http://www.compadre.org/STP/bulletinboard/Thread.cfm?ID=11286 Thu, 30 Jun 2011 09:44:23 EST http://www.compadre.org/STP/items/detail.cfm?ID=11286 http://www.compadre.org/STP/items/detail.cfm?ID=11283 Several Monte Carlo algorithms and applications that are useful for understanding the concepts of temperature and chemical potential are discussed. The authors then introduce a generalization of the demon algorithm that measures the chemical potential and is suitable for simulating systems with variable particle number. Thermo & Stat Mech/General http://www.compadre.org/STP/bulletinboard/Thread.cfm?ID=11283 Wed, 29 Jun 2011 16:50:39 EST http://www.compadre.org/STP/items/detail.cfm?ID=11283 http://www.compadre.org/STP/items/detail.cfm?ID=11284 The author discusses a geometrical phase transition involving percolation clusters. This continuous phase transition can be easily simulated on a computer and is analogous to the critical point in the Ising model. Thermo & Stat Mech/Phase Transitions/Continuous Transitions http://www.compadre.org/STP/bulletinboard/Thread.cfm?ID=11284 Wed, 29 Jun 2011 16:47:52 EST http://www.compadre.org/STP/items/detail.cfm?ID=11284 http://www.compadre.org/STP/items/detail.cfm?ID=11276 An review by Daniel ben-Avraham in the Journal of Statistical Physics of the textbook by Gould and Tobochnik. Thermo & Stat Mech/General http://www.compadre.org/STP/bulletinboard/Thread.cfm?ID=11276 Wed, 29 Jun 2011 16:33:03 EST http://www.compadre.org/STP/items/detail.cfm?ID=11276 http://www.compadre.org/STP/items/detail.cfm?ID=11282 The formation of crystalline and amorphous solids is described using a simple glass bead demonstration and a discussion of annealing and rapid quenching on a computer. Thermo & Stat Mech/Phase Transitions/First-Order Transitions http://www.compadre.org/STP/bulletinboard/Thread.cfm?ID=11282 Wed, 29 Jun 2011 10:29:11 EST http://www.compadre.org/STP/items/detail.cfm?ID=11282 http://www.compadre.org/STP/items/detail.cfm?ID=11281 The critical temperatures of the Ising model is obtained by considering the elementary cells of the square lattice in two dimensions and the cubic lattice in three dimensions. The cell configurations are divided into nondegenerate and degenerate classes. At the critical temperature of the infinite lattice the contributions of these two classes of are assumed to be equal. This conjecture reproduces the exact result for two dimensions and the numerical result for three-dimensional lattices. Although this conjecture is not exact, it gives insight into the nature of the transitions. Thermo & Stat Mech/Models/Ising Model http://www.compadre.org/STP/bulletinboard/Thread.cfm?ID=11281 Wed, 29 Jun 2011 10:09:25 EST http://www.compadre.org/STP/items/detail.cfm?ID=11281