Translation of Ludwig Boltzmann's Paper
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Canonical Ensemble
ME346A Introduction to Statistical Mechanics { Wei Cai { Stanford University { Win 2011 Handout 8. Canonical Ensemble January 26, 2011 Contents Outline • In this chapter, we will establish the equilibrium statistical distribution for systems maintained at a constant temperature T , through thermal contact with a heat bath. • The resulting distribution is also called Boltzmann's distribution. • The canonical distribution also leads to definition of the partition function and an expression for Helmholtz free energy, analogous to Boltzmann's Entropy formula. • We will study energy fluctuation at constant temperature, and witness another fluctuation- dissipation theorem (FDT) and finally establish the equivalence of micro canonical ensemble and canonical ensemble in the thermodynamic limit. (We first met a mani- festation of FDT in diffusion as Einstein's relation.) Reading Assignment: Reif x6.1-6.7, x6.10 1 1 Temperature For an isolated system, with fixed N { number of particles, V { volume, E { total energy, it is most conveniently described by the microcanonical (NVE) ensemble, which is a uniform distribution between two constant energy surfaces. const E ≤ H(fq g; fp g) ≤ E + ∆E ρ (fq g; fp g) = i i (1) mc i i 0 otherwise Statistical mechanics also provides the expression for entropy S(N; V; E) = kB ln Ω. In thermodynamics, S(N; V; E) can be transformed to a more convenient form (by Legendre transform) of Helmholtz free energy A(N; V; T ), which correspond to a system with constant N; V and temperature T . Q: Does the transformation from N; V; E to N; V; T have a meaning in statistical mechanics? A: The ensemble of systems all at constant N; V; T is called the canonical NVT ensemble. -
Magnetism, Magnetic Properties, Magnetochemistry
Magnetism, Magnetic Properties, Magnetochemistry 1 Magnetism All matter is electronic Positive/negative charges - bound by Coulombic forces Result of electric field E between charges, electric dipole Electric and magnetic fields = the electromagnetic interaction (Oersted, Maxwell) Electric field = electric +/ charges, electric dipole Magnetic field ??No source?? No magnetic charges, N-S No magnetic monopole Magnetic field = motion of electric charges (electric current, atomic motions) Magnetic dipole – magnetic moment = i A [A m2] 2 Electromagnetic Fields 3 Magnetism Magnetic field = motion of electric charges • Macro - electric current • Micro - spin + orbital momentum Ampère 1822 Poisson model Magnetic dipole – magnetic (dipole) moment [A m2] i A 4 Ampere model Magnetism Microscopic explanation of source of magnetism = Fundamental quantum magnets Unpaired electrons = spins (Bohr 1913) Atomic building blocks (protons, neutrons and electrons = fermions) possess an intrinsic magnetic moment Relativistic quantum theory (P. Dirac 1928) SPIN (quantum property ~ rotation of charged particles) Spin (½ for all fermions) gives rise to a magnetic moment 5 Atomic Motions of Electric Charges The origins for the magnetic moment of a free atom Motions of Electric Charges: 1) The spins of the electrons S. Unpaired spins give a paramagnetic contribution. Paired spins give a diamagnetic contribution. 2) The orbital angular momentum L of the electrons about the nucleus, degenerate orbitals, paramagnetic contribution. The change in the orbital moment -
Chapter 3. Second and Third Law of Thermodynamics
Chapter 3. Second and third law of thermodynamics Important Concepts Review Entropy; Gibbs Free Energy • Entropy (S) – definitions Law of Corresponding States (ch 1 notes) • Entropy changes in reversible and Reduced pressure, temperatures, volumes irreversible processes • Entropy of mixing of ideal gases • 2nd law of thermodynamics • 3rd law of thermodynamics Math • Free energy Numerical integration by computer • Maxwell relations (Trapezoidal integration • Dependence of free energy on P, V, T https://en.wikipedia.org/wiki/Trapezoidal_rule) • Thermodynamic functions of mixtures Properties of partial differential equations • Partial molar quantities and chemical Rules for inequalities potential Major Concept Review • Adiabats vs. isotherms p1V1 p2V2 • Sign convention for work and heat w done on c=C /R vm system, q supplied to system : + p1V1 p2V2 =Cp/CV w done by system, q removed from system : c c V1T1 V2T2 - • Joule-Thomson expansion (DH=0); • State variables depend on final & initial state; not Joule-Thomson coefficient, inversion path. temperature • Reversible change occurs in series of equilibrium V states T TT V P p • Adiabatic q = 0; Isothermal DT = 0 H CP • Equations of state for enthalpy, H and internal • Formation reaction; enthalpies of energy, U reaction, Hess’s Law; other changes D rxn H iD f Hi i T D rxn H Drxn Href DrxnCpdT Tref • Calorimetry Spontaneous and Nonspontaneous Changes First Law: when one form of energy is converted to another, the total energy in universe is conserved. • Does not give any other restriction on a process • But many processes have a natural direction Examples • gas expands into a vacuum; not the reverse • can burn paper; can't unburn paper • heat never flows spontaneously from cold to hot These changes are called nonspontaneous changes. -
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. -
Blackbody Radiation: (Vibrational Energies of Atoms in Solid Produce BB Radiation)
Independent study in physics The Thermodynamic Interaction of Light with Matter Mirna Alhanash Project in Physics Uppsala University Contents Abstract ................................................................................................................................................................................ 3 Introduction ......................................................................................................................................................................... 3 Blackbody Radiation: (vibrational energies of atoms in solid produce BB radiation) .................................... 4 Stefan-Boltzmann .............................................................................................................................................................. 6 Wien displacement law..................................................................................................................................................... 7 Photoelectric effect ......................................................................................................................................................... 12 Frequency dependence/Atom model & electron excitation .................................................................................. 12 Why we see colours ....................................................................................................................................................... 14 Optical properties of materials: .................................................................................................................................. -
Calculating the Configurational Entropy of a Landscape Mosaic
Landscape Ecol (2016) 31:481–489 DOI 10.1007/s10980-015-0305-2 PERSPECTIVE Calculating the configurational entropy of a landscape mosaic Samuel A. Cushman Received: 15 August 2014 / Accepted: 29 October 2015 / Published online: 7 November 2015 Ó Springer Science+Business Media Dordrecht (outside the USA) 2015 Abstract of classes and proportionality can be arranged (mi- Background Applications of entropy and the second crostates) that produce the observed amount of total law of thermodynamics in landscape ecology are rare edge (macrostate). and poorly developed. This is a fundamental limitation given the centrally important role the second law plays Keywords Entropy Á Landscape Á Configuration Á in all physical and biological processes. A critical first Composition Á Thermodynamics step to exploring the utility of thermodynamics in landscape ecology is to define the configurational entropy of a landscape mosaic. In this paper I attempt to link landscape ecology to the second law of Introduction thermodynamics and the entropy concept by showing how the configurational entropy of a landscape mosaic Entropy and the second law of thermodynamics are may be calculated. central organizing principles of nature, but are poorly Result I begin by drawing parallels between the developed and integrated in the landscape ecology configuration of a categorical landscape mosaic and literature (but see Li 2000, 2002; Vranken et al. 2014). the mixing of ideal gases. I propose that the idea of the Descriptions of landscape patterns, processes of thermodynamic microstate can be expressed as unique landscape change, and propagation of pattern-process configurations of a landscape mosaic, and posit that relationships across scale and through time are all the landscape metric Total Edge length is an effective governed and constrained by the second law of measure of configuration for purposes of calculating thermodynamics (Cushman 2015). -
Ludwig Boltzmann Was Born in Vienna, Austria. He Received His Early Education from a Private Tutor at Home
Ludwig Boltzmann (1844-1906) Ludwig Boltzmann was born in Vienna, Austria. He received his early education from a private tutor at home. In 1863 he entered the University of Vienna, and was awarded his doctorate in 1866. His thesis was on the kinetic theory of gases under the supervision of Josef Stefan. Boltzmann moved to the University of Graz in 1869 where he was appointed chair of the department of theoretical physics. He would move six more times, occupying chairs in mathematics and experimental physics. Boltzmann was one of the most highly regarded scientists, and universities wishing to increase their prestige would lure him to their institutions with high salaries and prestigious posts. Boltzmann himself was subject to mood swings and he joked that this was due to his being born on the night between Shrove Tuesday and Ash Wednesday (or between Carnival and Lent). Traveling and relocating would temporarily provide relief from his depression. He married Henriette von Aigentler in 1876. They had three daughters and two sons. Boltzmann is best known for pioneering the field of statistical mechanics. This work was done independently of J. Willard Gibbs (who never moved from his home in Connecticut). Together their theories connected the seemingly wide gap between the macroscopic properties and behavior of substances with the microscopic properties and behavior of atoms and molecules. Interestingly, the history of statistical mechanics begins with a mathematical prize at Cambridge in 1855 on the subject of evaluating the motions of Saturn’s rings. (Laplace had developed a mechanical theory of the rings concluding that their stability was due to irregularities in mass distribution.) The prize was won by James Clerk Maxwell who then went on to develop the theory that, without knowing the individual motions of particles (or molecules), it was possible to use their statistical behavior to calculate properties of a gas such as viscosity, collision rate, diffusion rate and the ability to conduct heat. -
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 -
Spin Statistics, Partition Functions and Network Entropy
This is a repository copy of Spin Statistics, Partition Functions and Network Entropy. White Rose Research Online URL for this paper: https://eprints.whiterose.ac.uk/118694/ Version: Accepted Version Article: Wang, Jianjia, Wilson, Richard Charles orcid.org/0000-0001-7265-3033 and Hancock, Edwin R orcid.org/0000-0003-4496-2028 (2017) Spin Statistics, Partition Functions and Network Entropy. Journal of Complex Networks. pp. 1-25. ISSN 2051-1329 https://doi.org/10.1093/comnet/cnx017 Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ IMA Journal of Complex Networks (2017)Page 1 of 25 doi:10.1093/comnet/xxx000 Spin Statistics, Partition Functions and Network Entropy JIANJIA WANG∗ Department of Computer Science,University of York, York, YO10 5DD, UK ∗[email protected] RICHARD C. WILSON Department of Computer Science,University of York, York, YO10 5DD, UK AND EDWIN R. HANCOCK Department of Computer Science,University of York, York, YO10 5DD, UK [Received on 2 May 2017] This paper explores the thermodynamic characterization of networks using the heat bath analogy when the energy states are occupied under different spin statistics, specified by a partition function. -
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. -
The Philosophy and Physics of Time Travel: the Possibility of Time Travel
University of Minnesota Morris Digital Well University of Minnesota Morris Digital Well Honors Capstone Projects Student Scholarship 2017 The Philosophy and Physics of Time Travel: The Possibility of Time Travel Ramitha Rupasinghe University of Minnesota, Morris, [email protected] Follow this and additional works at: https://digitalcommons.morris.umn.edu/honors Part of the Philosophy Commons, and the Physics Commons Recommended Citation Rupasinghe, Ramitha, "The Philosophy and Physics of Time Travel: The Possibility of Time Travel" (2017). Honors Capstone Projects. 1. https://digitalcommons.morris.umn.edu/honors/1 This Paper is brought to you for free and open access by the Student Scholarship at University of Minnesota Morris Digital Well. It has been accepted for inclusion in Honors Capstone Projects by an authorized administrator of University of Minnesota Morris Digital Well. For more information, please contact [email protected]. The Philosophy and Physics of Time Travel: The possibility of time travel Ramitha Rupasinghe IS 4994H - Honors Capstone Project Defense Panel – Pieranna Garavaso, Michael Korth, James Togeas University of Minnesota, Morris Spring 2017 1. Introduction Time is mysterious. Philosophers and scientists have pondered the question of what time might be for centuries and yet till this day, we don’t know what it is. Everyone talks about time, in fact, it’s the most common noun per the Oxford Dictionary. It’s in everything from history to music to culture. Despite time’s mysterious nature there are a lot of things that we can discuss in a logical manner. Time travel on the other hand is even more mysterious.