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A Simple Method to Estimate Entropy and Free Energy of Atmospheric Gases from Their Action
Article A Simple Method to Estimate Entropy and Free Energy of Atmospheric Gases from Their Action Ivan Kennedy 1,2,*, Harold Geering 2, Michael Rose 3 and Angus Crossan 2 1 Sydney Institute of Agriculture, University of Sydney, NSW 2006, Australia 2 QuickTest Technologies, PO Box 6285 North Ryde, NSW 2113, Australia; [email protected] (H.G.); [email protected] (A.C.) 3 NSW Department of Primary Industries, Wollongbar NSW 2447, Australia; [email protected] * Correspondence: [email protected]; Tel.: + 61-4-0794-9622 Received: 23 March 2019; Accepted: 26 April 2019; Published: 1 May 2019 Abstract: A convenient practical model for accurately estimating the total entropy (ΣSi) of atmospheric gases based on physical action is proposed. This realistic approach is fully consistent with statistical mechanics, but reinterprets its partition functions as measures of translational, rotational, and vibrational action or quantum states, to estimate the entropy. With all kinds of molecular action expressed as logarithmic functions, the total heat required for warming a chemical system from 0 K (ΣSiT) to a given temperature and pressure can be computed, yielding results identical with published experimental third law values of entropy. All thermodynamic properties of gases including entropy, enthalpy, Gibbs energy, and Helmholtz energy are directly estimated using simple algorithms based on simple molecular and physical properties, without resource to tables of standard values; both free energies are measures of quantum field states and of minimal statistical degeneracy, decreasing with temperature and declining density. We propose that this more realistic approach has heuristic value for thermodynamic computation of atmospheric profiles, based on steady state heat flows equilibrating with gravity. -
Statistics of a Free Single Quantum Particle at a Finite
STATISTICS OF A FREE SINGLE QUANTUM PARTICLE AT A FINITE TEMPERATURE JIAN-PING PENG Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, China Abstract We present a model to study the statistics of a single structureless quantum particle freely moving in a space at a finite temperature. It is shown that the quantum particle feels the temperature and can exchange energy with its environment in the form of heat transfer. The underlying mechanism is diffraction at the edge of the wave front of its matter wave. Expressions of energy and entropy of the particle are obtained for the irreversible process. Keywords: Quantum particle at a finite temperature, Thermodynamics of a single quantum particle PACS: 05.30.-d, 02.70.Rr 1 Quantum mechanics is the theoretical framework that describes phenomena on the microscopic level and is exact at zero temperature. The fundamental statistical character in quantum mechanics, due to the Heisenberg uncertainty relation, is unrelated to temperature. On the other hand, temperature is generally believed to have no microscopic meaning and can only be conceived at the macroscopic level. For instance, one can define the energy of a single quantum particle, but one can not ascribe a temperature to it. However, it is physically meaningful to place a single quantum particle in a box or let it move in a space where temperature is well-defined. This raises the well-known question: How a single quantum particle feels the temperature and what is the consequence? The question is particular important and interesting, since experimental techniques in recent years have improved to such an extent that direct measurement of electron dynamics is possible.1,2,3 It should also closely related to the question on the applicability of the thermodynamics to small systems on the nanometer scale.4 We present here a model to study the behavior of a structureless quantum particle moving freely in a space at a nonzero temperature. -
LECTURES on MATHEMATICAL STATISTICAL MECHANICS S. Adams
ISSN 0070-7414 Sgr´ıbhinn´ıInstiti´uid Ard-L´einnBhaile´ Atha´ Cliath Sraith. A. Uimh 30 Communications of the Dublin Institute for Advanced Studies Series A (Theoretical Physics), No. 30 LECTURES ON MATHEMATICAL STATISTICAL MECHANICS By S. Adams DUBLIN Institi´uid Ard-L´einnBhaile´ Atha´ Cliath Dublin Institute for Advanced Studies 2006 Contents 1 Introduction 1 2 Ergodic theory 2 2.1 Microscopic dynamics and time averages . .2 2.2 Boltzmann's heuristics and ergodic hypothesis . .8 2.3 Formal Response: Birkhoff and von Neumann ergodic theories9 2.4 Microcanonical measure . 13 3 Entropy 16 3.1 Probabilistic view on Boltzmann's entropy . 16 3.2 Shannon's entropy . 17 4 The Gibbs ensembles 20 4.1 The canonical Gibbs ensemble . 20 4.2 The Gibbs paradox . 26 4.3 The grandcanonical ensemble . 27 4.4 The "orthodicity problem" . 31 5 The Thermodynamic limit 33 5.1 Definition . 33 5.2 Thermodynamic function: Free energy . 37 5.3 Equivalence of ensembles . 42 6 Gibbs measures 44 6.1 Definition . 44 6.2 The one-dimensional Ising model . 47 6.3 Symmetry and symmetry breaking . 51 6.4 The Ising ferromagnet in two dimensions . 52 6.5 Extreme Gibbs measures . 57 6.6 Uniqueness . 58 6.7 Ergodicity . 60 7 A variational characterisation of Gibbs measures 62 8 Large deviations theory 68 8.1 Motivation . 68 8.2 Definition . 70 8.3 Some results for Gibbs measures . 72 i 9 Models 73 9.1 Lattice Gases . 74 9.2 Magnetic Models . 75 9.3 Curie-Weiss model . 77 9.4 Continuous Ising model . -
Entropy and H Theorem the Mathematical Legacy of Ludwig Boltzmann
ENTROPY AND H THEOREM THE MATHEMATICAL LEGACY OF LUDWIG BOLTZMANN Cambridge/Newton Institute, 15 November 2010 C´edric Villani University of Lyon & Institut Henri Poincar´e(Paris) Cutting-edge physics at the end of nineteenth century Long-time behavior of a (dilute) classical gas Take many (say 1020) small hard balls, bouncing against each other, in a box Let the gas evolve according to Newton’s equations Prediction by Maxwell and Boltzmann The distribution function is asymptotically Gaussian v 2 f(t, x, v) a exp | | as t ≃ − 2T → ∞ Based on four major conceptual advances 1865-1875 Major modelling advance: Boltzmann equation • Major mathematical advance: the statistical entropy • Major physical advance: macroscopic irreversibility • Major PDE advance: qualitative functional study • Let us review these advances = journey around centennial scientific problems ⇒ The Boltzmann equation Models rarefied gases (Maxwell 1865, Boltzmann 1872) f(t, x, v) : density of particles in (x, v) space at time t f(t, x, v) dxdv = fraction of mass in dxdv The Boltzmann equation (without boundaries) Unknown = time-dependent distribution f(t, x, v): ∂f 3 ∂f + v = Q(f,f) = ∂t i ∂x Xi=1 i ′ ′ B(v v∗,σ) f(t, x, v )f(t, x, v∗) f(t, x, v)f(t, x, v∗) dv∗ dσ R3 2 − − Z v∗ ZS h i The Boltzmann equation (without boundaries) Unknown = time-dependent distribution f(t, x, v): ∂f 3 ∂f + v = Q(f,f) = ∂t i ∂x Xi=1 i ′ ′ B(v v∗,σ) f(t, x, v )f(t, x, v∗) f(t, x, v)f(t, x, v∗) dv∗ dσ R3 2 − − Z v∗ ZS h i The Boltzmann equation (without boundaries) Unknown = time-dependent distribution -
A Molecular Modeler's Guide to Statistical Mechanics
A Molecular Modeler’s Guide to Statistical Mechanics Course notes for BIOE575 Daniel A. Beard Department of Bioengineering University of Washington Box 3552255 [email protected] (206) 685 9891 April 11, 2001 Contents 1 Basic Principles and the Microcanonical Ensemble 2 1.1 Classical Laws of Motion . 2 1.2 Ensembles and Thermodynamics . 3 1.2.1 An Ensembles of Particles . 3 1.2.2 Microscopic Thermodynamics . 4 1.2.3 Formalism for Classical Systems . 7 1.3 Example Problem: Classical Ideal Gas . 8 1.4 Example Problem: Quantum Ideal Gas . 10 2 Canonical Ensemble and Equipartition 15 2.1 The Canonical Distribution . 15 2.1.1 A Derivation . 15 2.1.2 Another Derivation . 16 2.1.3 One More Derivation . 17 2.2 More Thermodynamics . 19 2.3 Formalism for Classical Systems . 20 2.4 Equipartition . 20 2.5 Example Problem: Harmonic Oscillators and Blackbody Radiation . 21 2.5.1 Classical Oscillator . 22 2.5.2 Quantum Oscillator . 22 2.5.3 Blackbody Radiation . 23 2.6 Example Application: Poisson-Boltzmann Theory . 24 2.7 Brief Introduction to the Grand Canonical Ensemble . 25 3 Brownian Motion, Fokker-Planck Equations, and the Fluctuation-Dissipation Theo- rem 27 3.1 One-Dimensional Langevin Equation and Fluctuation- Dissipation Theorem . 27 3.2 Fokker-Planck Equation . 29 3.3 Brownian Motion of Several Particles . 30 3.4 Fluctuation-Dissipation and Brownian Dynamics . 32 1 Chapter 1 Basic Principles and the Microcanonical Ensemble The first part of this course will consist of an introduction to the basic principles of statistical mechanics (or statistical physics) which is the set of theoretical techniques used to understand microscopic systems and how microscopic behavior is reflected on the macroscopic scale. -
Entropy: from the Boltzmann Equation to the Maxwell Boltzmann Distribution
Entropy: From the Boltzmann equation to the Maxwell Boltzmann distribution A formula to relate entropy to probability Often it is a lot more useful to think about entropy in terms of the probability with which different states are occupied. Lets see if we can describe entropy as a function of the probability distribution between different states. N! WN particles = n1!n2!....nt! stirling (N e)N N N W = = N particles n1 n2 nt n1 n2 nt (n1 e) (n2 e) ....(nt e) n1 n2 ...nt with N pi = ni 1 W = N particles n1 n2 nt p1 p2 ...pt takeln t lnWN particles = "#ni ln pi i=1 divide N particles t lnW1particle = "# pi ln pi i=1 times k t k lnW1particle = "k# pi ln pi = S1particle i=1 and t t S = "N k p ln p = "R p ln p NA A # i # i i i=1 i=1 ! Think about how this equation behaves for a moment. If any one of the states has a probability of occurring equal to 1, then the ln pi of that state is 0 and the probability of all the other states has to be 0 (the sum over all probabilities has to be 1). So the entropy of such a system is 0. This is exactly what we would have expected. In our coin flipping case there was only one way to be all heads and the W of that configuration was 1. Also, and this is not so obvious, the maximal entropy will result if all states are equally populated (if you want to see a mathematical proof of this check out Ken Dill’s book on page 85). -
Physics 357 – Thermal Physics - Spring 2005
Physics 357 – Thermal Physics - Spring 2005 Contact Info MTTH 9-9:50. Science 255 Doug Juers, Science 244, 527-5229, [email protected] Office Hours: Mon 10-noon, Wed 9-10 and by appointment or whenever I’m around. Description Thermal physics involves the study of systems with lots (1023) of particles. Since only a two-body problem can be solved analytically, you can imagine we will look at these systems somewhat differently from simple systems in mechanics. Rather than try to predict individual trajectories of individual particles we will consider things on a larger scale. Probability and statistics will play an important role in thinking about what can happen. There are two general methods of dealing with such systems – a top down approach and a bottom up approach. In the top down approach (thermodynamics) one doesn’t even really consider that there are particles there, and instead thinks about things in terms of quantities like temperature heat, energy, work, entropy and enthalpy. In the bottom up approach (statistical mechanics) one considers in detail the interactions between individual particles. Building up from there yields impressive predictions of bulk properties of different materials. In this course we will cover both thermodynamics and statistical mechanics. The first part of the term will emphasize the former, while the second part of the term will emphasize the latter. Both topics have important applications in many fields, including solid-state physics, fluids, geology, chemistry, biology, astrophysics, cosmology. The great thing about thermal physics is that in large part it’s about trying to deal with reality. -
Boltzmann Equation II: Binary Collisions
Physics 127b: Statistical Mechanics Boltzmann Equation II: Binary Collisions Binary collisions in a classical gas Scattering out Scattering in v'1 v'2 v 1 v 2 R v 2 v 1 v'2 v'1 Center of V' Mass Frame V θ θ b sc sc R V V' Figure 1: Binary collisions in a gas: top—lab frame; bottom—centre of mass frame Binary collisions in a gas are very similar, except that the scattering is off another molecule. An individual scattering process is, of course, simplest to describe in the center of mass frame in terms of the relative E velocity V =Ev1 −Ev2. However the center of mass frame is different for different collisions, so we must keep track of the results in the lab frame, and this makes the calculation rather intricate. I will indicate the main ideas here, and refer you to Reif or Landau and Lifshitz for precise discussions. Lets first set things up in the lab frame. Again we consider the pair of scattering in and scattering out processes that are space-time inverses, and so have identical cross sections. We can write abstractly for the scattering out from velocity vE1 due to collisions with molecules with all velocities vE2, which will clearly be proportional to the number f (vE1) of molecules at vE1 (which we write as f1—sorry, not the same notation as in the previous sections where f1 denoted the deviation of f from the equilibrium distribution!) and the 3 3 number f2d v2 ≡ f(vE2)d v2 in each velocity volume element, and then we must integrate over all possible vE0 vE0 outgoing velocities 1 and 2 ZZZ df (vE ) 1 =− w(vE0 , vE0 ;Ev , vE )f f d3v d3v0 d3v0 . -
Boltzmann Equation
Boltzmann Equation ● Velocity distribution functions of particles ● Derivation of Boltzmann Equation Ludwig Eduard Boltzmann (February 20, 1844 - September 5, 1906), an Austrian physicist famous for the invention of statistical mechanics. Born in Vienna, Austria-Hungary, he committed suicide in 1906 by hanging himself while on holiday in Duino near Trieste in Italy. Distribution Function (probability density function) Random variable y is distributed with the probability density function f(y) if for any interval [a b] the probability of a<y<b is equal to b P=∫ f ydy a f(y) is always non-negative ∫ f ydy=1 Velocity space Axes u,v,w in velocity space v dv have the same directions as dv axes x,y,z in physical du dw u space. Each molecule can be v represented in velocity space by the point defined by its velocity vector v with components (u,v,w) w Velocity distribution function Consider a sample of gas that is homogeneous in physical space and contains N identical molecules. Velocity distribution function is defined by d N =Nf vd ud v d w (1) where dN is the number of molecules in the sample with velocity components (ui,vi,wi) such that u<ui<u+du, v<vi<v+dv, w<wi<w+dw dv = dudvdw is a volume element in the velocity space. Consequently, dN is the number of molecules in velocity space element dv. Functional statement if often omitted, so f(v) is designated as f Phase space distribution function Macroscopic properties of the flow are functions of position and time, so the distribution function depends on position and time as well as velocity. -
Nonlinear Electrostatics. the Poisson-Boltzmann Equation
Nonlinear Electrostatics. The Poisson-Boltzmann Equation C. G. Gray* and P. J. Stiles# *Department of Physics, University of Guelph, Guelph, ON N1G2W1, Canada ([email protected]) #Department of Molecular Sciences, Macquarie University, NSW 2109, Australia ([email protected]) The description of a conducting medium in thermal equilibrium, such as an electrolyte solution or a plasma, involves nonlinear electrostatics, a subject rarely discussed in the standard electricity and magnetism textbooks. We consider in detail the case of the electrostatic double layer formed by an electrolyte solution near a uniformly charged wall, and we use mean-field or Poisson-Boltzmann (PB) theory to calculate the mean electrostatic potential and the mean ion concentrations, as functions of distance from the wall. PB theory is developed from the Gibbs variational principle for thermal equilibrium of minimizing the system free energy. We clarify the key issue of which free energy (Helmholtz, Gibbs, grand, …) should be used in the Gibbs principle; this turns out to depend not only on the specified conditions in the bulk electrolyte solution (e.g., fixed volume or fixed pressure), but also on the specified surface conditions, such as fixed surface charge or fixed surface potential. Despite its nonlinearity the PB equation for the mean electrostatic potential can be solved analytically for planar or wall geometry, and we present analytic solutions for both a full electrolyte, and for an ionic solution which contains only counterions, i.e. ions of sign opposite to that of the wall charge. This latter case has some novel features. We also use the free energy to discuss the inter-wall forces which arise when the two parallel charged walls are sufficiently close to permit their double layers to overlap. -
Physics, M.S. 1
Physics, M.S. 1 PHYSICS, M.S. DEPARTMENT OVERVIEW The Department of Physics has a strong tradition of graduate study and research in astrophysics; atomic, molecular, and optical physics; condensed matter physics; high energy and particle physics; plasma physics; quantum computing; and string theory. There are many facilities for carrying out world-class research (https://www.physics.wisc.edu/ research/areas/). We have a large professional staff: 45 full-time faculty (https://www.physics.wisc.edu/people/staff/) members, affiliated faculty members holding joint appointments with other departments, scientists, senior scientists, and postdocs. There are over 175 graduate students in the department who come from many countries around the world. More complete information on the graduate program, the faculty, and research groups is available at the department website (http:// www.physics.wisc.edu). Research specialties include: THEORETICAL PHYSICS Astrophysics; atomic, molecular, and optical physics; condensed matter physics; cosmology; elementary particle physics; nuclear physics; phenomenology; plasmas and fusion; quantum computing; statistical and thermal physics; string theory. EXPERIMENTAL PHYSICS Astrophysics; atomic, molecular, and optical physics; biophysics; condensed matter physics; cosmology; elementary particle physics; neutrino physics; experimental studies of superconductors; medical physics; nuclear physics; plasma physics; quantum computing; spectroscopy. M.S. DEGREES The department offers the master science degree in physics, with two named options: Research and Quantum Computing. The M.S. Physics-Research option (http://guide.wisc.edu/graduate/physics/ physics-ms/physics-research-ms/) is non-admitting, meaning it is only available to students pursuing their Ph.D. The M.S. Physics-Quantum Computing option (http://guide.wisc.edu/graduate/physics/physics-ms/ physics-quantum-computing-ms/) (MSPQC Program) is a professional master's program in an accelerated format designed to be completed in one calendar year.. -
Physics Ph.D
THE UNIVERSITY OF VERMONT PHYSICS PH.D. PHYSICS PH.D. consists of a Ph.D. dissertation proposal given after the start of a dissertation research project. All students must meet the Requirements for the Doctor of Philosophy Degree (http:// Requirements for Advancement to Candidacy for the catalogue.uvm.edu/graduate/degreerequirements/ Degree of Doctor of Philosophy requirementsforthedoctorofphilosophydegree/). Successful completion of all required courses and the comprehensive exam. OVERVIEW The Department of Physics offers research opportunities in theoretical and experimental condensed matter physics, astronomy and astrophysics, and soft condensed matter physics and biophysics. SPECIFIC REQUIREMENTS Requirements for Admission to Graduate Studies for the Degree of Doctor of Philosophy Undergraduate majors in physics are considered for admission to the program. Satisfactory scores on the Graduate Record Examination (general) are required. Minimum Degree Requirements 75 credits, including: 6 Core Graduate Courses PHYS 301 Mathematical Physics 3 PHYS 311 Advanced Dynamics 3 PHYS 313 Electromagnetic Theory 3 PHYS 323 Contemporary Physics 3 PHYS 362 Quantum Mechanics II 3 PHYS 365 Statistical Mechanics 3 All of these courses must be completed with a grade B or better within the first 2 years of graduate study. To accommodate the needs of the specific subfields in physics such as astrophysics, biological physics, condensed- matter physics and materials physics, 3 elective courses (9 credits) have to be chosen to fulfill the breadth requirement with a grade of B or higher. Elective courses must be completed within the first 3 years of the program, as the fourth year (and beyond if needed) should be dedicated to progress towards the Ph.D.