http://www.compadre.org/STP/ The latest material additions to the Statistical and Thermal Physics. en-US Copyright 2013, ComPADRE.org stp@compadre.org stp@compadre.org Tue, 08 Jan 2013 14:07:05 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=11563 The Boltzmann Distribution From A Microcanonical Ensemble Model allows students and instructors to explore why the Boltzmann distribution has its characteristic exponential shape. In this model, particles have only one degree of freedom–the energy to move in one dimension. Further, the density of accessible states is chosen to be uniform –i.e. each state is equally probable. Starting with a system of N=2 particles, you are able to generate simple empirical results for the distribution of energies among the individual particles. Then you can observe how this distribution changes with increasing N, gradually approaching the Boltzmann distribution. The Boltzmann Distribution From A Microcanonical Ensemble Model was developed using the Easy Java Simulations (EJS) modeling tool. It is distributed as a ready-to-run (compiled) Java archive. Double clicking the ejs_fmu_BoltzmannDistribution_PartitioningEnergy.jar file will run the program if Java is installed. You can modify this simulation if you have EJS installed by right-clicking within the map and selecting "Open Ejs Model" from the pop-up menu item. Thermo & Stat Mech/Ensembles/Boltzmann Distribution http://www.compadre.org/STP/bulletinboard/Thread.cfm?ID=11563 Tue, 08 Jan 2013 14:07:05 EST http://www.compadre.org/STP/items/detail.cfm?ID=11563 http://www.compadre.org/STP/items/detail.cfm?ID=11230 An upper division biophysics course containing assignments that are suitable for teaching in an interdisciplinary manner. The assignments contain over 50 simulations written in SciPy/python. There are detailed instructions that come with the assignments. Thermo & Stat Mech/General http://www.compadre.org/STP/bulletinboard/Thread.cfm?ID=11230 Fri, 04 Jan 2013 21:56:43 EST http://www.compadre.org/STP/items/detail.cfm?ID=11230 http://www.compadre.org/STP/items/detail.cfm?ID=9947 The Binomial Coefficient model displays the number of ways k objects can be chosen from among n objects when order is irrelevant.   This number is known as a binomial coefficient and can be used to predict the the flipping of n coins with equal probability of heads and tails. The Binomial Coefficient model was created using the Easy Java Simulations (EJS) modeling tool. It is distributed as a ready-to-run (compiled) Java archive. Double clicking the ejs_stp_BinomialCoefficient.jar file will run the program if Java is installed. Thermo & Stat Mech/Probability/Binomial Distribution http://www.compadre.org/STP/bulletinboard/Thread.cfm?ID=9947 Fri, 04 Jan 2013 21:45:14 EST http://www.compadre.org/STP/items/detail.cfm?ID=9947 http://www.compadre.org/STP/items/detail.cfm?ID=12435 The Second Law of Thermodynamics and Boltzmann's Statistics model performs successive simulations of an isolated assembly of vibrating molecules modeled as quantum harmonic oscillators. The objective is to reach, by means of molecular random collisions, the thermodynamic equilibrium detected through the entropy evolution and the approach to Boltzmann's distribution. When the simulation is run, each successive simulation consists of random collisions (the simulation number is indicated in the middle yellow field). The reduced entropy (S/k) evolution is displayed at the top panel after every 50 collisions as a function of the total number of collisions. The Second Law of Thermodynamics and Boltzmann's Statistics model was developed using the Easy Java Simulations (EJS) modeling tool. It is distributed as a ready-to-run (compiled) Java archive. Double clicking the jar file will run the program if Java is installed. You can modify this simulation if you have EJS installed by right-clicking within the map and selecting "Open Ejs Model" from the pop-up menu item. Thermo & Stat Mech/Second and Third Law/Entropy http://www.compadre.org/STP/bulletinboard/Thread.cfm?ID=12435 Fri, 04 Jan 2013 21:39:50 EST http://www.compadre.org/STP/items/detail.cfm?ID=12435 http://www.compadre.org/STP/items/detail.cfm?ID=11715 Describes an experiment to acquaint students with physical illustrations of concepts in Fourier analysis using a simple experimental setup involving wavelike behavior. In addition, students measure the speed of propagation of thermal oscillations, analyze the heat equation and measure the thermal diffusivity of the material under observation. The objectives of the experiment are: 1. to understand the basis of heat flow and recognize heat conduction as a diffusive process, 2. to learn about solutions of the heat equation, 3. to decompose an oscillation into its harmonics, 4. to observe different harmonics and how they damp with different rates, and 5. to estimate the thermal diffusivity of a metal. Thermo & Stat Mech/Kinetics and Dynamics/Thermal Conductivity http://www.compadre.org/STP/bulletinboard/Thread.cfm?ID=11715 Fri, 04 Jan 2013 21:38:21 EST http://www.compadre.org/STP/items/detail.cfm?ID=11715 http://www.compadre.org/STP/items/detail.cfm?ID=12147 Molecular Workbench (MW) provides visual, interactive computational experiments for teaching and learning science. Concepts of temperature, phase changes, and energy transfer shown in molecular dynamics simulations of atoms interacting with Leonard-Jones potential. Thermo & Stat Mech/General http://www.compadre.org/STP/bulletinboard/Thread.cfm?ID=12147 Mon, 02 Jul 2012 03:01:10 EST http://www.compadre.org/STP/items/detail.cfm?ID=12147 http://www.compadre.org/STP/items/detail.cfm?ID=3461 This website contains a large collection of student activities which focus on the molecular basis of common phenomena. Each activity provides guided interactive explorations of atomic and molecular computational models, many with accompanying teacher guides and student assessments. A primary goal of the Molecular Logic project is to build and publicize a library of "deeply digital" examples that support technology-based curriculum materials. The collection covers a wide range of topics in physics, biology, and chemistry. The models are all freely accessible. Users must register for additional free access to capture data and store student work products. General Physics/Collections http://www.compadre.org/STP/bulletinboard/Thread.cfm?ID=3461 Mon, 02 Jul 2012 02:45:53 EST http://www.compadre.org/STP/items/detail.cfm?ID=3461 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