Thermal and Statistical Physics Simulations

• Harvey Gould of Clark University
• Lynna Spornick of Johns Hopkins University
• Jan Tobochnik of Kalamazoo College

• ENGDRV written by Lynna Spornick, is a driver for four programs that provide an introduction to the thermodynamics of engines.
  • ENGINE (Design Your Own Engine) lets the user design an engine by specifying the processes (adiabatic, isobaric, isochoric (constant volume), and isothermic) in the engine's cycle, the engine type (reversible or irreversible), and the gas type (Helium, Argon, Nitrogen, or Steam). The thermodynamic properties (heat exchange, work done, and change in Internal Energy) for each process and the engine's efficiency are computed.

  • DIESEL, OTTO, and WANKEL provide animations of each of these types of engine. Plots of the Temperature versus Entropy and the Pressure versus Volume for the cycles are shown with the engine's current thermodynamic conditions indicated.

• PROBDRV written by Lynna Spornick, is a driver program for: subprograms GALTON, POISSON, and TWOD that provide an introduction to Probability and subprograms KAC and STADIUM that provide an introduction to Statistics.
  • GALTON (A Galton Board) models either a traditional Galton Board or a customized Galton Board with traps, reflecting, and/or absorbing walls. GALTON demonstrates the Binominal and Normal Distributions, the Laws of Probability, and the Central Limit Theorem.

  • POISSON (Poisson Probability Distribution in Nuclear Decay), written by Lynna Spornick, uses the decay of radioactive atoms to describe the Poisson and the Exponential Distributions.

  • TWOD (Two-Dimensional Random Walk) models a random walk in two dimensions. A "drunk", taking equal length steps, is required to walk either on a grid or on a plane. TWOD demonstrates the joint probability of two independent processes, the Binominal Distribution, and the Rayleigh Distribution.

  • KAC (A Kac Ring) uses a Kac Ring to demonstrate that large mechanical systems, whose Equations of Motion are solvable and which obey time reversal and have a Poincare Cycle, can also be described by Statistical Models.

  • STADIUM (The Stadium Model) uses a Stadium Model to demonstrate that there exists mechanical systems whose Equations of Motion are solvable but whose motion is not predictable because of the system's Chaotic Nature.

• ISING (Ising Model In One and Two Dimensions), written by Harvey Gould, allows the user to explore the static and dynamic properties of the one- and two-dimensional Ising model using four different Monte Carlo algorithms and three different ensembles. The choice of the Metropolis algorithm allows the user to study the Ising model at constant temperature and external magnetic field. The orientation of the spins is shown on the screen as well as the evolution of the total energy or magnetization. The mean energy, magnetization, heat capacity, and susceptibility are monitored as a function of the number of configurations that are sampled. Other computed quantities include the equilibrium-averaged energy and magnetization autocorrelation functions and the energy histogram. Important physical concepts that can be studied with the aid of the program include the Boltzmann probability, the qualitative behavior of systems near critical points, critical exponents, the renormalization group, and critical slowing down. Other algorithms that can be chosen by the user correspond to spin exchange dynamics (constant magnetization), constant energy (the demon algorithm), and single cluster Wolff dynamics. The latter is particularly useful for generating equilibrium configurations at the critical point.

• MOLDYN (Molecular Dynamics), written by Harvey Gould, allows the user to simulate a dense gas, liquid, or solid in two dimensions using either molecular dynamics (constant energy, constant volume) or Monte Carlo (constant temperature, constant volume) methods. Both hard disks and the Lennard-Jones interaction can be chosen. The trajectories for the particles are shown as the system evolves. Physical quantities of interest that are monitored include the pressure, temperature, heat capacity, mean square displacement, the distribution of the speeds and velocities, and the pair correlation function. Important physical concepts that can be studied with the aid of the program include the Maxwell-Boltzmann probability distribution, fluctuations, equation of state, correlations, and the importance of chaotic mixing.

• FLUIDS (Thermodynamics of Fluids), written by Jan Tobochnik, allows the user to explore the fluid (gas and liquid) phases diagrams for the van der Waals model and water. The user chooses four phase diagrams from among the following choices: PT, Pv, vT, uT, sT, uv, and sv, where P is the pressure, T is the temperature, v is the specific volume, S is the specific entropy, and u is the specific internal energy. The program reads in the coexistence table for the van der Waals model. Given v and u, any thermodynamic quantity can be calculated. For the van der waals model thermodynamic quantities also can be calculated from the other thermodynamic state variables. The user can draw a straight line path in one phase diagram and see how this path looks in the other phase diagrams. The user can also extract all important thermodynamic data at any point in a phase diagram.

• QMGAS1 (Quantum Mechanical Gas - Part 1), written by Jan Tobochnik, does the numerical calculations necessary to solve for the thermodynamic properties of quantum ideal gases including photons in blackbody radiation, ideal bosons, phonons in the Debye theory, non-interacting fermions, and the classical limits of these systems. The user chooses the type of statistics (Bose-Einstein, Fermi-Dirac, or Maxwell-Boltzmann), the dimension of space, the form of the dispersion relation (restricted to simple powers), whether or not the particles have a non-zero chemical potential, and whether or not there is a Debye cutoff. The program then allows the user to build up a table of thermodynamic data including the energy, specific hear, and chemical potential as a function of temperature. This data and various distribution functions and the density of states can be plotted.

• QMGAS2 (Quantum Mechanical Gas - Part 2), written by Jan Tobochnik implements a Monte Carlo simulation of a finite number of quantum particles fluctuating between various states in a finite k-space (k is the wavevector). The program orders the possible energy states into an energy level diagram and then allows particles to move from one state to another according to the usual Boltzmann probability distribution. Bosons are restricted so that they may not pass through each other on the energy level diagram; fermions are further restricted so that no two fermions may be in the same state; classical particles have no restrictions. In this way indistinguishability is correctly implemented for bosons and fermions. The user chooses the type of particle, the number of particles, the size and dimension of k-space, and the temperature. During the simulation the user sees a representation of the state occupancy and plots of the average energy, the instantaneous energy, and the distribution of energy among the states, also, shown are results for the average energy, specific heat, and the occupancy of the ground state.