Entropy: from the Boltzmann Equation to the Maxwell Boltzmann Distribution
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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 -
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. -
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. -
Physical Biology
Physical Biology Kevin Cahill Department of Physics and Astronomy University of New Mexico, Albuquerque, NM 87131 i For Ginette, Mike, Sean, Peter, Mia, James, Dylan, and Christopher Contents Preface page v 1 Atoms and Molecules 1 1.1 Atoms 1 1.2 Bonds between atoms 1 1.3 Oxidation and reduction 5 1.4 Molecules in cells 6 1.5 Sugars 6 1.6 Hydrocarbons, fatty acids, and fats 8 1.7 Nucleotides 10 1.8 Amino acids 15 1.9 Proteins 19 Exercises 20 2 Metabolism 23 2.1 Glycolysis 23 2.2 The citric-acid cycle 23 3 Molecular devices 27 3.1 Transport 27 4 Probability and statistics 28 4.1 Mean and variance 28 4.2 The binomial distribution 29 4.3 Random walks 32 4.4 Diffusion as a random walk 33 4.5 Continuous diffusion 34 4.6 The Poisson distribution 36 4.7 The gaussian distribution 37 4.8 The error function erf 39 4.9 The Maxwell-Boltzmann distribution 40 Contents iii 4.10 Characteristic functions 41 4.11 Central limit theorem 42 Exercises 45 5 Statistical Mechanics 47 5.1 Probability and the Boltzmann distribution 47 5.2 Equilibrium and the Boltzmann distribution 49 5.3 Entropy 53 5.4 Temperature and entropy 54 5.5 The partition function 55 5.6 The Helmholtz free energy 57 5.7 The Gibbs free energy 58 Exercises 59 6 Chemical Reactions 61 7 Liquids 65 7.1 Diffusion 65 7.2 Langevin's Theory of Brownian Motion 67 7.3 The Einstein-Nernst relation 71 7.4 The Poisson-Boltzmann equation 72 7.5 Reynolds number 73 7.6 Fluid mechanics 76 8 Cell structure 77 8.1 Organelles 77 8.2 Membranes 78 8.3 Membrane potentials 80 8.4 Potassium leak channels 82 8.5 The Debye-H¨uckel equation 83 8.6 The Poisson-Boltzmann equation in one dimension 87 8.7 Neurons 87 Exercises 89 9 Polymers 90 9.1 Freely jointed polymers 90 9.2 Entropic force 91 10 Mechanics 93 11 Some Experimental Methods 94 11.1 Time-dependent perturbation theory 94 11.2 FRET 96 11.3 Fluorescence 103 11.4 Some mathematics 107 iv Contents References 109 Index 111 Preface Most books on biophysics hide the physics so as to appeal to a wide audi- ence of students. -
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. -
Chapter 6. the Ising Model on a Square Lattice
Chapter 6. The Ising model on a square lattice 6.1 The basic Ising model The Ising model is a minimal model to capture the spontaneous emergence of macroscopic ferromagnetism from the interaction of individual microscopic magnetic dipole moments. Consider a two-dimensional system of spin 1=2 magnetic dipoles arrayed on a square L L lattice. At each lattice point .i; j /, the dipole can assume only one of two spin states, i;j 1. The dipoles interact with an externally applied constant magnetic field B, and the energyD˙ of a dipole interacting with the external field, Uext, is given by U .i;j / Hi;j ext D where H is a non-dimensionalized version of B. The energy of each spin (dipole) is lower when it is aligned with the external magnetic field. The dipoles also interact with their nearest neighbors. Due to quantum effects the energy of two spins in a ferromagnetic material is lower when they are aligned with each other1. As a model of this, it is assumed in the Ising model that the interaction energy for the dipole at .i; j / is given by Uint.i;j / Ji;j .i 1;j i 1;j i;j 1 i;j 1/ D C C C C C where J is a non-dimensionalized coupling strength. Summing over all sites, we get the total energy for a configuration given by E L L J L L U./ H i;j i;j .i 1;j i 1;j i;j 1 i;j 1/: E D 2 C C C C C i 1 j 1 i 1 j 1 XD XD XD XD The second sum is divided by two to account for each interaction being counted twice. -
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.