The Phases of Hard Sphere Systems
|
|
- Cory Dixon
- 6 years ago
- Views:
Transcription
1 The Phases of Hard Sphere Systems Tutorial for Spring 2015 ICERM workshop Crystals, Quasicrystals and Random Networks Veit Elser Cornell Department of Physics
2 outline: order by disorder constant temperature & pressure ensemble phases and coexistence ordered-phase nucleation infinite pressure limit what is known in low dimensions the wilds of higher dimensions
3
4
5
6 eliminating time: ergodic hypothesis sphere centers: x 1,x 2,...,x N arbitrary property: (x 1,x 2,...,x N )
7 eliminating time: ergodic hypothesis sphere centers: x 1,x 2,...,x N arbitrary property: (x 1,x 2,...,x N ) time average: = 1 T Z T 0 dt (x 1 (t),...,x N (t))
8 eliminating time: ergodic hypothesis sphere centers: x 1,x 2,...,x N arbitrary property: (x 1,x 2,...,x N ) time average: configuration average: = 1 T Z h i = Z T 0 dt (x 1 (t),...,x N (t)) dµ (x 1,...,x N )
9 eliminating time: ergodic hypothesis sphere centers: x 1,x 2,...,x N arbitrary property: (x 1,x 2,...,x N ) time average: configuration average: = 1 T Z h i = Z T 0 dt (x 1 (t),...,x N (t)) dµ (x 1,...,x N ) ergodic hypothesis: = h i
10 Gibbs measure: dµ = d D x 1 d D x N e H 0 (x 1,...,x N ) domain: (x 1,...,x N ) 2 A N A R D box
11 Gibbs measure: dµ = d D x 1 d D x N e H 0 (x 1,...,x N ) domain: (x 1,...,x N ) 2 A N A R D box absolute temperature: sphere diameter: 1/ = k B T a
12 Gibbs measure: dµ = d D x 1 d D x N e H 0 (x 1,...,x N ) domain: (x 1,...,x N ) 2 A N A R D box absolute temperature: sphere diameter: 1/ = k B T a Hamiltonian: H 0 (x 1,...,x N )= 0 if 8 i 6= j kxi x j k >a 1 otherwise
13 constant temperature & pressure ensemble environment V pv = work performed on constant pressure environment in expanding box to volume V T, p H(x 1,...,x N ; V )=H 0 (x 1,...,x N )+pv
14 dimensionless variables & parameters x i = a x i V = a D Ṽ
15 dimensionless variables & parameters x i = a x i V = a D Ṽ pv = p Ṽ p =(k B T/a D ) p
16 rescaled constant T & p ensemble measure: dµ(p) =d D x 1 d D x N dv e H 0 pv
17 rescaled constant T & p ensemble measure: dµ(p) =d D x 1 d D x N dv e H 0 pv configurations: (x 1,...,x N ) 2 A(V ) N A(V )=R D /(V 1/D Z) D (periodic box) 8 i 6= j kx i x j k > 1
18 rescaled constant T & p ensemble measure: dµ(p) =d D x 1 d D x N dv e H 0 pv configurations: (x 1,...,x N ) 2 A(V ) N A(V )=R D /(V 1/D Z) D (periodic box) 8 i 6= j kx i x j k > 1 dimensionless pressure (parameter): p specific volume (property): v = hv i/n
19
20
21 p =2.00 v =1.73
22 p =2.00 v =1.73
23 p = v =1.02
24 p = v =1.02
25 108 spheres (D = 3) v =1/ p 2
26 packing fraction: = B D(1/2) v
27 quiz: hard sphere phase diagram p gas crystal p crystal gas p crystal gas T T T
28 quiz: hard sphere phase diagram p gas crystal p crystal gas p crystal gas T T T p = p (k B /a 3 ) T p 10 a 4Å (krypton) p (k B /a 3 ) 20 atm/k
29 thermal equilibrium Here we are, trapped in the amber of the moment. Kurt Vonnegut configuration sampling via Markov chains, small transitions Markov sampling is not unlike time evolution diffusion process: slow for large systems
30 approaching equilibrium 108 spheres ṗ = p t =ṗ>0 > 0
31 approaching equilibrium 108 spheres ṗ = p 1 2 p t =ṗ>0 > 0
32 quiz: between phases Suppose we sample hard spheres (3D) at fixed volume, corresponding to an intermediate packing fraction (0.52), what should we expect to see in a typical configuration? A. gas phase B. crystal phase C. quantum superposition of gas and crystal D. none of the above
33 quiz: between phases Suppose we sample hard spheres (3D) at fixed volume, corresponding to an intermediate packing fraction (0.52), what should we expect to see in a typical configuration? A. gas phase B. crystal phase C. quantum superposition of gas and crystal D. none of the above
34 phase coexistence At the coexistence pressure, gas and crystal subsystems are in equilibrium with each other. We can accommodate a range of volumes by changing the relative fractions of the two phases. p = { gas gas 1 =0.494 crystal 1 < < 2 crystal 2 =0.545
35 thermodynamics number of microstates: Z(p) = Z dµ(p) = Z dµ(0)e pv
36 thermodynamics number of microstates: Z(p) = Z dµ(p) = Z dµ(0)e pv entropy per sphere: s(p) = k B N log Z(p)
37 thermodynamics number of microstates: Z(p) = Z dµ(p) = Z dµ(0)e pv entropy per sphere: s(p) = k B N log Z(p) free energy per sphere: F (p) = Ts(p)
38 thermodynamics number of microstates: Z(p) = Z dµ(p) = Z dµ(0)e pv entropy per sphere: s(p) = k B N log Z(p) free energy per sphere: F (p) = Ts(p) identity: v = 1 N hv i = 1 N d dp log Z(p) = 1 k B ds dp
39 s gas (p )=s cryst (p ) s gas s(p) s cryst p v gas v = k 1 B ds dp v v cryst p
40 phase equilibrium: the ice-9 problem suppose, young man, that one Marine had with him a tiny capsule containing a seed of ice-nine, a new way for the atoms of water to stack and lock, to freeze. If that Marine threw that seed into the nearest puddle? Kurt Vonnegut (Cat s Cradle) Phase coexistence implies there is an entropic penalty, a negative contribution from the interface, proportional to its area. The interfacial penalty severely inhibits the spontaneous formation of the ordered phase, especially near the coexistence pressure. critical nucleus
41 size of critical nucleus Assume interface-surface entropy is isotropic, so the critical nucleus is spherical:
42 size of critical nucleus Assume interface-surface entropy is isotropic, so the critical nucleus is spherical: s nuc =(s cryst s gas ) V nuc v A nuc
43 size of critical nucleus Assume interface-surface entropy is isotropic, so the critical nucleus is spherical: s nuc =(s cryst s gas ) V nuc v A nuc s cryst (p + p) s gas (p + p) =k B ( v cryst + v gas ) p = k B ( v) p>0
44 size of critical nucleus Assume interface-surface entropy is isotropic, so the critical nucleus is spherical: s nuc =(s cryst s gas ) V nuc v A nuc s cryst (p + p) s gas (p + p) =k B ( v cryst + v gas ) p = k B ( v) p>0 V nuc =(4 /3)R 3 A nuc =4 R 2
45 size of critical nucleus Assume interface-surface entropy is isotropic, so the critical nucleus is spherical: s nuc =(s cryst s gas ) V nuc v A nuc s cryst (p + p) s gas (p + p) =k B ( v cryst + v gas ) p = k B ( v) p>0 V nuc =(4 /3)R 3 A nuc =4 R 2 s nuc R c
46 s nuc (R c )=0 ) R c = 3( /k B )(v/ v) 1 p exponentially small nucleation rate: e V nuc/v 0
47 s nuc (R c )=0 ) R c = 3( /k B )(v/ v) 1 p exponentially small nucleation rate: e V nuc/v 0 The critical nucleus would in any case have to be at least as large to contain most of a kissing sphere configuration, thus making the spontaneous nucleation rate decay exponentially with dimension.
48 infinite pressure limit lim p!1 k 1 B ds dp = v min
49 infinite pressure limit lim p!1 k 1 B ds dp = v min s(p)/k B v min p + C
50 infinite pressure limit lim p!1 k 1 B ds dp = v min s(p)/k B v min p + C Z(p) e N(v min p+c)
51 infinite pressure limit lim p!1 k 1 B ds dp = v min s(p)/k B v min p + C Z(p) e N(v min p+c) In three dimensions we know there are many sphere packings stacking sequences of hexagonal layers with exactly the same vmin. The constant C depends on the stacking sequence.
52 free volume: disks
53 free volume: disks Allowed positions of central disk when surrounding disks are held fixed.
54 free volume: spheres =0.45 cubic hexagonal AB hexagonal AC
55 hexagonal stacking B A C hexagonal close packing (hcp) : ABAB face-centered cubic (fcc) : ABCABC
56 densest & most probable sphere packing lim s(p)/k B = C fcc C hcp p!1 N = 1000 : prob(fcc) prob(hcp) 3.2 fcc and hcp stackings are the extreme cases; other stackings have intermediate probabilities.
57 hard sphere phases in other dimensions The statistical ensemble, through phase transitions, detects any kind of order. Good packings assert themselves at lower packing fractions. Different dimensions have qualitatively different phase behaviors. what we know so far
58 D =1 only one phase (gas) exercise: calculate s(p) in your head
59 D =2 If you had to devote your life to one dimension, this would be the one. three phases: gas, hexatic, crystal gas: 0 < < gas-hexatic coexistence: < < 0.714, p =9.185 hexatic: crystal: < < < hexatic-crystal transition is Kosterlitz*-Thouless logarithmic fluctuations in crystal (Mermin-Wagner) gas-hexatic coexistence resolved only recently (Michael Engel, et al.) * Brown Faculty
60 2 <D<7 As in three dimensions, two phases: gas and crystal. Symmetry of crystal phase consistent with densest lattice. Ice-9 problem avoided by introducing tethers to crystal sites whose strengths are eventually reduced to zero ( Einstein-crystal method ). Possible ordered phases limited to chosen crystal types. Interesting behavior of coexistence pressure: D gas cryst max p type D D D E6 source: J.A. van Meel, PhD thesis
61 hard spheres in higher dimensions: three possible universes
62 Leech s universe Every dimension has a unique and efficiently constructible densest packing of spheres. At low density the gas phase maximizes the entropy. There is only one other phase, at higher density, where spheres fluctuate about the centers of the densest sphere packing (appropriately scaled). Classic order by disorder!
63 The Stillinger-Torquato* universe It is unreasonable to expect to find lattices optimized for packing spheres in high dimensions. Argument: For a lattice in dimension D the number of parameters in its specification grows as a polynomial in D. But the number of spheres that might impinge on any given sphere is of order the kissing number and grows exponentially with D. Conjecture: In high dimensions and large volumes V, there is a proliferation of sphere packings in V very close in density to the densest (much greater in number than defected crystal packings). Because of this proliferation, the densest packings are actually disordered. Since the lack of order in the densest limit is qualitatively no different from the lack of order in the gas, there is only one phase! * S. Torquato and F. H. Stillinger, Experimental Mathematics, Vol. 15 (2006)
64 Conway s universe Mathematical invention knows no bounds, and natural phenomena never fail to surprise. Perhaps two dimensions was not an anomaly: multiple phases of increasing degrees of order may lurk in higher dimensions. Testimonial: unbiased kissing configuration search in ten dimensions.
65 The densest known sphere packing in 10 dimensions, P10c, has center density δ = 5/128. For comparison, D10 has center density δ = 2/128. P10c, a non-lattice, has maximum kissing number 372. Output of kissing number search with constraint satisfaction algorithm:
66 The densest known sphere packing in 10 dimensions, P10c, has center density δ = 5/128. For comparison, D10 has center density δ = 2/128. P10c, a non-lattice, has maximum kissing number 372. Output of kissing number search with constraint satisfaction algorithm: 372 spheres 374 spheres cos Θ disordered cos Θ ordered
67 The densest known sphere packing in 10 dimensions, P10c, has center density δ = 5/128. For comparison, D10 has center density δ = 2/128. P10c, a non-lattice, has maximum kissing number 372. Output of kissing number search with constraint satisfaction algorithm: 372 spheres 374 spheres cos Θ disordered cos Θ ordered cos 2 {0, (3 ± p 3)/12, 1/2} (378 spheres in complete configuration)
68 The new kissing configuration extends to a quasicrystal in 10 dimensions with center density δ = 4/128! Sphere centers projected into quasiperiodic plane; five kinds of contacting spheres. Quasiperiodic tiling of squares, triangles and rhombi. V. Elser and S. Gravel, Discrete Comput. Geom (2010)
Heterogenous Nucleation in Hard Spheres Systems
University of Luxembourg, Softmatter Theory Group May, 2012 Table of contents 1 2 3 Motivation Event Driven Molecular Dynamics Time Driven MD simulation vs. Event Driven MD simulation V(r) = { if r < σ
More information5 Topological defects and textures in ordered media
5 Topological defects and textures in ordered media In this chapter we consider how to classify topological defects and textures in ordered media. We give here only a very short account of the method following
More informationPrinciples of Equilibrium Statistical Mechanics
Debashish Chowdhury, Dietrich Stauffer Principles of Equilibrium Statistical Mechanics WILEY-VCH Weinheim New York Chichester Brisbane Singapore Toronto Table of Contents Part I: THERMOSTATICS 1 1 BASIC
More information1 The fundamental equation of equilibrium statistical mechanics. 3 General overview on the method of ensembles 10
Contents EQUILIBRIUM STATISTICAL MECHANICS 1 The fundamental equation of equilibrium statistical mechanics 2 2 Conjugate representations 6 3 General overview on the method of ensembles 10 4 The relation
More information6.730 Physics for Solid State Applications
6.730 Physics for Solid State Applications Lecture 25: Chemical Potential and Equilibrium Outline Microstates and Counting System and Reservoir Microstates Constants in Equilibrium Temperature & Chemical
More informationCalculating the Free Energy of Nearly Jammed Hard-Particle Packings Using Molecular Dynamics
Calculating the Free Energy of Nearly Jammed Hard-Particle Packings Using Molecular Dynamics Aleksandar Donev, 1, 2 Frank H. Stillinger, 3 1, 4, 3, 5, and Salvatore Torquato 1 Program in Applied and Computational
More informationcollisions inelastic, energy is dissipated and not conserved
LECTURE 1 - Introduction to Granular Materials Jamming, Random Close Packing, The Isostatic State Granular materials particles only interact when they touch hard cores: rigid, incompressible, particles
More informationPHYS 352 Homework 2 Solutions
PHYS 352 Homework 2 Solutions Aaron Mowitz (, 2, and 3) and Nachi Stern (4 and 5) Problem The purpose of doing a Legendre transform is to change a function of one or more variables into a function of variables
More informationEquilibrium Phase Behavior and Maximally. Random Jammed State of Truncated Tetrahedra
Equilibrium Phase Behavior and Maximally Random Jammed State of Truncated Tetrahedra Duyu Chen,, Yang Jiao, and Salvatore Torquato,,,,, Department of Chemistry, Princeton University, Princeton, New Jersey
More informationPhysics Qual - Statistical Mechanics ( Fall 2016) I. Describe what is meant by: (a) A quasi-static process (b) The second law of thermodynamics (c) A throttling process and the function that is conserved
More informationLecture 6 - Bonding in Crystals
Lecture 6 onding in Crystals inding in Crystals (Kittel Ch. 3) inding of atoms to form crystals A crystal is a repeated array of atoms Why do they form? What are characteristic bonding mechanisms? How
More informationPhysics 541: Condensed Matter Physics
Physics 541: Condensed Matter Physics In-class Midterm Exam Wednesday, October 26, 2011 / 14:00 15:20 / CCIS 4-285 Student s Name: Instructions There are 23 questions. You should attempt all of them. Mark
More informationStatistical Thermodynamics and Monte-Carlo Evgenii B. Rudnyi and Jan G. Korvink IMTEK Albert Ludwig University Freiburg, Germany
Statistical Thermodynamics and Monte-Carlo Evgenii B. Rudnyi and Jan G. Korvink IMTEK Albert Ludwig University Freiburg, Germany Preliminaries Learning Goals From Micro to Macro Statistical Mechanics (Statistical
More informationDisordered Hyperuniformity: Liquid-like Behaviour in Structural Solids, A New Phase of Matter?
Disordered Hyperuniformity: Liquid-like Behaviour in Structural Solids, A New Phase of Matter? Kabir Ramola Martin Fisher School of Physics, Brandeis University August 19, 2016 Kabir Ramola Disordered
More informationThe Solid State. Phase diagrams Crystals and symmetry Unit cells and packing Types of solid
The Solid State Phase diagrams Crystals and symmetry Unit cells and packing Types of solid Learning objectives Apply phase diagrams to prediction of phase behaviour Describe distinguishing features of
More informationMany-Body Localization. Geoffrey Ji
Many-Body Localization Geoffrey Ji Outline Aside: Quantum thermalization; ETH Single-particle (Anderson) localization Many-body localization Some phenomenology (l-bit model) Numerics & Experiments Thermalization
More informationIntroduction Statistical Thermodynamics. Monday, January 6, 14
Introduction Statistical Thermodynamics 1 Molecular Simulations Molecular dynamics: solve equations of motion Monte Carlo: importance sampling r 1 r 2 r n MD MC r 1 r 2 2 r n 2 3 3 4 4 Questions How can
More information(# = %(& )(* +,(- Closed system, well-defined energy (or e.g. E± E/2): Microcanonical ensemble
Recall from before: Internal energy (or Entropy): &, *, - (# = %(& )(* +,(- Closed system, well-defined energy (or e.g. E± E/2): Microcanonical ensemble & = /01Ω maximized Ω: fundamental statistical quantity
More informationThermal and Statistical Physics Department Exam Last updated November 4, L π
Thermal and Statistical Physics Department Exam Last updated November 4, 013 1. a. Define the chemical potential µ. Show that two systems are in diffusive equilibrium if µ 1 =µ. You may start with F =
More informationThermodynamics is the only science about which I am firmly convinced that, within the framework of the applicability of its basic principles, it will
Thermodynamics is the only science about which I am firmly convinced that, within the framework of the applicability of its basic principles, it will never be overthrown - Albert Einstein OFP Chapter 11
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Statistical Physics I Spring Term 2013 Notes on the Microcanonical Ensemble
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.044 Statistical Physics I Spring Term 2013 Notes on the Microcanonical Ensemble The object of this endeavor is to impose a simple probability
More informationG : Statistical Mechanics
G25.2651: Statistical Mechanics Notes for Lecture 1 Defining statistical mechanics: Statistical Mechanics provies the connection between microscopic motion of individual atoms of matter and macroscopically
More informationPhysics 172H Modern Mechanics
Physics 172H Modern Mechanics Instructor: Dr. Mark Haugan Office: PHYS 282 haugan@purdue.edu TAs: Alex Kryzwda John Lorenz akryzwda@purdue.edu jdlorenz@purdue.edu Lecture 22: Matter & Interactions, Ch.
More informationEntropic Crystal-Crystal Transitions of Brownian Squares K. Zhao, R. Bruinsma, and T.G. Mason
Entropic Crystal-Crystal Transitions of Brownian Squares K. Zhao, R. Bruinsma, and T.G. Mason This supplementary material contains the following sections: image processing methods, measurements of Brownian
More informationNENG 301 Week 8 Unary Heterogeneous Systems (DeHoff, Chap. 7, Chap )
NENG 301 Week 8 Unary Heterogeneous Systems (DeHoff, Chap. 7, Chap. 5.3-5.4) Learning objectives for Chapter 7 At the end of this chapter you will be able to: Understand the general features of a unary
More informationPhase Transitions in Physics and Computer Science. Cristopher Moore University of New Mexico and the Santa Fe Institute
Phase Transitions in Physics and Computer Science Cristopher Moore University of New Mexico and the Santa Fe Institute Magnetism When cold enough, Iron will stay magnetized, and even magnetize spontaneously
More informationIntroduction to Phase Transitions in Statistical Physics and Field Theory
Introduction to Phase Transitions in Statistical Physics and Field Theory Motivation Basic Concepts and Facts about Phase Transitions: Phase Transitions in Fluids and Magnets Thermodynamics and Statistical
More informationStatistical Mechanics
42 My God, He Plays Dice! Statistical Mechanics Statistical Mechanics 43 Statistical Mechanics Statistical mechanics and thermodynamics are nineteenthcentury classical physics, but they contain the seeds
More informationDisordered Energy Minimizing Configurations
. p. /4 Disordered Energy Minimizing Configurations Salvatore Torquato Department of Chemistry, Department of Physics, Princeton Institute for the Science and Technology of Materials, and Program in Applied
More informationPhysical Biochemistry. Kwan Hee Lee, Ph.D. Handong Global University
Physical Biochemistry Kwan Hee Lee, Ph.D. Handong Global University Week 3 CHAPTER 2 The Second Law: Entropy of the Universe increases What is entropy Definition: measure of disorder The greater the disorder,
More informationPrinciples of Condensed matter physics
Principles of Condensed matter physics P. M. CHAIKIN Princeton University T. C. LUBENSKY University of Pennsylvania CAMBRIDGE UNIVERSITY PRESS Preface xvii 1 Overview 1 1.1 Condensed matter physics 1 1.2
More informationSpontaneous Symmetry Breaking
Spontaneous Symmetry Breaking Second order phase transitions are generally associated with spontaneous symmetry breaking associated with an appropriate order parameter. Identifying symmetry of the order
More informationPhysics 212: Statistical mechanics II Lecture XI
Physics 212: Statistical mechanics II Lecture XI The main result of the last lecture was a calculation of the averaged magnetization in mean-field theory in Fourier space when the spin at the origin is
More information3d Crystallization. FCC structures. Florian Theil. Macroscopic Modeling of Materials with Fine Structure May Warwick University
FCC structures 1 Mathematics Institute Warwick University Macroscopic Modeling of Materials with Fine Structure May 2011 Outline 1 2 Outline 1 2 Extremum principles Crystalline structure is the minimizer
More informationTomato Packing and Lettuce-Based Crypto
Tomato Packing and Lettuce-Based Crypto A. S. Mosunov and L. A. Ruiz-Lopez University of Waterloo Joint PM & C&O Colloquium March 30th, 2017 Spring in Waterloo Love is everywhere... Picture from https://avatanplus.com/files/resources/mid/
More informationRam Seshadri MRL 2031, x6129, These notes complement chapter 6 of Anderson, Leaver, Leevers and Rawlings
Crystals, packings etc. Ram Seshadri MRL 2031, x6129, seshadri@mrl.ucsb.edu These notes complement chapter 6 of Anderson, Leaver, Leevers and Rawlings The unit cell and its propagation Materials usually
More informationIntroduction to Solid State Physics or the study of physical properties of matter in a solid phase
Introduction to Solid State Physics or the study of physical properties of matter in a solid phase Prof. Germar Hoffmann 1. Crystal Structures 2. Reciprocal Lattice 3. Crystal Binding and Elastic Constants
More informationMonte Carlo. Lecture 15 4/9/18. Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky
Monte Carlo Lecture 15 4/9/18 1 Sampling with dynamics In Molecular Dynamics we simulate evolution of a system over time according to Newton s equations, conserving energy Averages (thermodynamic properties)
More informationStructure and Dynamics : An Atomic View of Materials
Structure and Dynamics : An Atomic View of Materials MARTIN T. DOVE Department ofearth Sciences University of Cambridge OXFORD UNIVERSITY PRESS Contents 1 Introduction 1 1.1 Observations 1 1.1.1 Microscopic
More informationReview of First and Second Law of Thermodynamics
Review of First and Second Law of Thermodynamics Reading Problems 4-1 4-4 4-32, 4-36, 4-87, 4-246 5-2 5-4, 5.7 6-1 6-13 6-122, 6-127, 6-130 Definitions SYSTEM: any specified collection of matter under
More information3.012 PS Issued: Fall 2003 Graded problems due:
3.012 PS 4 3.012 Issued: 10.07.03 Fall 2003 Graded problems due: 10.15.03 Graded problems: 1. Planes and directions. Consider a 2-dimensional lattice defined by translations T 1 and T 2. a. Is the direction
More informationDilatancy Transition in a Granular Model. David Aristoff and Charles Radin * Mathematics Department, University of Texas, Austin, TX 78712
Dilatancy Transition in a Granular Model by David Aristoff and Charles Radin * Mathematics Department, University of Texas, Austin, TX 78712 Abstract We introduce a model of granular matter and use a stress
More informationCollective Effects. Equilibrium and Nonequilibrium Physics
1 Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 2, 24 March 2006 1 Collective Effects in Equilibrium and Nonequilibrium Physics Website: http://cncs.bnu.edu.cn/mccross/course/ Caltech
More informationarxiv: v1 [cond-mat.soft] 20 Jun 2008
Accurate determination of crystal structures based on averaged local bond order parameters Wolfgang Lechner and Christoph Dellago Faculty of hysics, University of Vienna, Boltzmanngasse, 19 Vienna, Austria
More informationCHAPTER 4 Physical Transformations of Pure Substances.
I. Generalities. CHAPTER 4 Physical Transformations of Pure Substances. A. Definitions: 1. A phase of a substance is a form of matter that is uniform throughout in chemical composition and physical state.
More informationPhase Transitions in Condensed Matter Spontaneous Symmetry Breaking and Universality. Hans-Henning Klauss. Institut für Festkörperphysik TU Dresden
Phase Transitions in Condensed Matter Spontaneous Symmetry Breaking and Universality Hans-Henning Klauss Institut für Festkörperphysik TU Dresden 1 References [1] Stephen Blundell, Magnetism in Condensed
More informationTopological defects and its role in the phase transition of a dense defect system
Topological defects and its role in the phase transition of a dense defect system Suman Sinha * and Soumen Kumar Roy Depatrment of Physics, Jadavpur University Kolkata- 70003, India Abstract Monte Carlo
More informationChapter 3. Entropy, temperature, and the microcanonical partition function: how to calculate results with statistical mechanics.
Chapter 3. Entropy, temperature, and the microcanonical partition function: how to calculate results with statistical mechanics. The goal of equilibrium statistical mechanics is to calculate the density
More informationVIII. Phase Transformations. Lecture 38: Nucleation and Spinodal Decomposition
VIII. Phase Transformations Lecture 38: Nucleation and Spinodal Decomposition MIT Student In this lecture we will study the onset of phase transformation for phases that differ only in their equilibrium
More informationCollective Effects. Equilibrium and Nonequilibrium Physics
Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March 2006 Collective Effects in Equilibrium and Nonequilibrium Physics Website: http://cncs.bnu.edu.cn/mccross/course/ Caltech
More informationEnergy-driven pattern formation. crystals
: Optimal crystals 1 Mathematics Institute Warwick University Wien October 2014 Polymorphism It is amazing that even complex molecules like proteins crystallize without significant interference. The anti-aids
More informationCHEM-UA 652: Thermodynamics and Kinetics
1 CHEM-UA 652: Thermodynamics and Kinetics Notes for Lecture 2 I. THE IDEAL GAS LAW In the last lecture, we discussed the Maxwell-Boltzmann velocity and speed distribution functions for an ideal gas. Remember
More informationGeometrical Frustration: Hard Spheres vs Cylinders Patrick Charbonneau
Geometrical Frustration: Hard Spheres vs Cylinders Patrick Charbonneau Prologue: (First) Glass Problem Credit: Patrick Charbonneau, 2012 Berthier and Ediger, Physics Today (2016) Crystal Nucleation Radius
More informationA Brief Introduction to Statistical Mechanics
A Brief Introduction to Statistical Mechanics E. J. Maginn, J. K. Shah Department of Chemical and Biomolecular Engineering University of Notre Dame Notre Dame, IN 46556 USA Monte Carlo Workshop Universidade
More informationAn Outline of (Classical) Statistical Mechanics and Related Concepts in Machine Learning
An Outline of (Classical) Statistical Mechanics and Related Concepts in Machine Learning Chang Liu Tsinghua University June 1, 2016 1 / 22 What is covered What is Statistical mechanics developed for? What
More informationEinstein s early work on Statistical Mechanics
Einstein s early work on Statistical Mechanics A prelude to the Marvelous Year Luca Peliti M. A. and H. Chooljan Member Simons Center for Systems Biology Institute for Advanced Study Princeton (USA) November
More informationSuggestions for Further Reading
Contents Preface viii 1 From Microscopic to Macroscopic Behavior 1 1.1 Introduction........................................ 1 1.2 Some Qualitative Observations............................. 2 1.3 Doing
More informationPrestress stability. Lecture VI. Session on Granular Matter Institut Henri Poincaré. R. Connelly Cornell University Department of Mathematics
Prestress stability Lecture VI Session on Granular Matter Institut Henri Poincaré R. Connelly Cornell University Department of Mathematics 1 Potential functions How is the stability of a structure determined
More informationThe Gibbs Phase Rule F = 2 + C - P
The Gibbs Phase Rule The phase rule allows one to determine the number of degrees of freedom (F) or variance of a chemical system. This is useful for interpreting phase diagrams. F = 2 + C - P Where F
More informationWhat is Classical Molecular Dynamics?
What is Classical Molecular Dynamics? Simulation of explicit particles (atoms, ions,... ) Particles interact via relatively simple analytical potential functions Newton s equations of motion are integrated
More informationChapter 9: Statistical Mechanics
Chapter 9: Statistical Mechanics Chapter 9: Statistical Mechanics...111 9.1 Introduction...111 9.2 Statistical Mechanics...113 9.2.1 The Hamiltonian...113 9.2.2 Phase Space...114 9.2.3 Trajectories and
More informationErgodicity,non-ergodic processes and aging processes
Ergodicity,non-ergodic processes and aging processes By Amir Golan Outline: A. Ensemble and time averages - Definition of ergodicity B. A short overvie of aging phenomena C. Examples: 1. Simple-glass and
More informationActivity 5&6: Metals and Hexagonal Close-Packing
Chemistry 150 Name(s): Activity 5&6: Metals and Hexagonal Close-Packing Metals are chemicals characterized by high thermal and electrical conductivity, malleability and ductility. Atoms are the smallest
More informationCase study: molecular dynamics of solvent diffusion in polymers
Course MP3 Lecture 11 29/11/2006 Case study: molecular dynamics of solvent diffusion in polymers A real-life research example to illustrate the use of molecular dynamics Dr James Elliott 11.1 Research
More informationClassical Statistical Mechanics: Part 1
Classical Statistical Mechanics: Part 1 January 16, 2013 Classical Mechanics 1-Dimensional system with 1 particle of mass m Newton s equations of motion for position x(t) and momentum p(t): ẋ(t) dx p =
More informationG : Statistical Mechanics Notes for Lecture 3 I. MICROCANONICAL ENSEMBLE: CONDITIONS FOR THERMAL EQUILIBRIUM Consider bringing two systems into
G25.2651: Statistical Mechanics Notes for Lecture 3 I. MICROCANONICAL ENSEMBLE: CONDITIONS FOR THERMAL EQUILIBRIUM Consider bringing two systems into thermal contact. By thermal contact, we mean that the
More information1. Introductory Examples
1. Introductory Examples We introduce the concept of the deterministic and stochastic simulation methods. Two problems are provided to explain the methods: the percolation problem, providing an example
More informationChapter 2. Atomic Packing
Chapter 2. Atomic Packing Contents 2-1. Packing of directional bonding atoms 2-2. Packing of indirectional bonding in same size atoms 2-3. Packing of indirectional bonding in different size atoms 2-4.
More informationarxiv:quant-ph/ v2 24 Dec 2003
Quantum Entanglement in Heisenberg Antiferromagnets V. Subrahmanyam Department of Physics, Indian Institute of Technology, Kanpur, India. arxiv:quant-ph/0309004 v2 24 Dec 2003 Entanglement sharing among
More informationOSU Physics Department Comprehensive Examination #115
1 OSU Physics Department Comprehensive Examination #115 Monday, January 7 and Tuesday, January 8, 2013 Winter 2013 Comprehensive Examination PART 1, Monday, January 7, 9:00am General Instructions This
More informationParamagnetic phases of Kagome lattice quantum Ising models p.1/16
Paramagnetic phases of Kagome lattice quantum Ising models Predrag Nikolić In collaboration with T. Senthil Massachusetts Institute of Technology Paramagnetic phases of Kagome lattice quantum Ising models
More informationLecture 9 Examples and Problems
Lecture 9 Examples and Problems Counting microstates of combined systems Volume exchange between systems Definition of Entropy and its role in equilibrium The second law of thermodynamics Statistics of
More informationPhysics 607 Exam 2. ( ) = 1, Γ( z +1) = zγ( z) x n e x2 dx = 1. e x2
Physics 607 Exam Please be well-organized, and show all significant steps clearly in all problems. You are graded on your work, so please do not just write down answers with no explanation! Do all your
More informationStructure of the First and Second Neighbor Shells of Water: Quantitative Relation with Translational and Orientational Order.
Structure of the First and Second Neighbor Shells of Water: Quantitative Relation with Translational and Orientational Order Zhenyu Yan, Sergey V. Buldyrev,, Pradeep Kumar, Nicolas Giovambattista 3, Pablo
More informationChem 728 Introduction to Solid Surfaces
Chem 728 Introduction to Solid Surfaces Solids: hard; fracture; not compressible; molecules close to each other Liquids: molecules mobile, but quite close to each other Gases: molecules very mobile; compressible
More informationFlip dynamics on canonical cut and project tilings
Flip dynamics on canonical cut and project tilings Thomas Fernique CNRS & Univ. Paris 13 M2 Pavages ENS Lyon November 5, 2015 Outline 1 Random tilings 2 Random sampling 3 Mixing time 4 Slow cooling Outline
More informationFermi gas model. Introduction to Nuclear Science. Simon Fraser University Spring NUCS 342 February 2, 2011
Fermi gas model Introduction to Nuclear Science Simon Fraser University Spring 2011 NUCS 342 February 2, 2011 NUCS 342 (Lecture 9) February 2, 2011 1 / 34 Outline 1 Bosons and fermions NUCS 342 (Lecture
More informationSTATISTICAL PHYSICS. Statics, Dynamics and Renormalization. Leo P Kadanoff. Departments of Physics & Mathematics University of Chicago
STATISTICAL PHYSICS Statics, Dynamics and Renormalization Leo P Kadanoff Departments of Physics & Mathematics University of Chicago \o * World Scientific Singapore»New Jersey London»HongKong Contents Introduction
More informationRemoving the mystery of entropy and thermodynamics. Part 3
Removing the mystery of entropy and thermodynamics. Part 3 arvey S. Leff a,b Physics Department Reed College, Portland, Oregon USA August 3, 20 Introduction In Part 3 of this five-part article, [, 2] simple
More informationMS212 Thermodynamics of Materials ( 소재열역학의이해 ) Lecture Note: Chapter 7
2017 Spring Semester MS212 Thermodynamics of Materials ( 소재열역학의이해 ) Lecture Note: Chapter 7 Byungha Shin ( 신병하 ) Dept. of MSE, KAIST Largely based on lecture notes of Prof. Hyuck-Mo Lee and Prof. WooChul
More informationLocalization I: General considerations, one-parameter scaling
PHYS598PTD A.J.Leggett 2013 Lecture 4 Localization I: General considerations 1 Localization I: General considerations, one-parameter scaling Traditionally, two mechanisms for localization of electron states
More informationChapter 2 Ensemble Theory in Statistical Physics: Free Energy Potential
Chapter Ensemble Theory in Statistical Physics: Free Energy Potential Abstract In this chapter, we discuss the basic formalism of statistical physics Also, we consider in detail the concept of the free
More information3.320 Lecture 18 (4/12/05)
3.320 Lecture 18 (4/12/05) Monte Carlo Simulation II and free energies Figure by MIT OCW. General Statistical Mechanics References D. Chandler, Introduction to Modern Statistical Mechanics D.A. McQuarrie,
More informationPHASE TRANSITIONS IN SOFT MATTER SYSTEMS
OUTLINE: Topic D. PHASE TRANSITIONS IN SOFT MATTER SYSTEMS Definition of a phase Classification of phase transitions Thermodynamics of mixing (gases, polymers, etc.) Mean-field approaches in the spirit
More informationNEWS ABOUT SYMMETRIESIN PHYSICS ISCMNS 12th Workshop
NEWS ABOUT SYMMETRIESIN PHYSICS ISCMNS 12th Workshop Jean-François Geneste Jenny D. Vinko 2 3 Let there be light! Said the Lord 4 And the light was! 5 In the beginning of humankind there was 6 The cave!
More informationCrystallographic Point Groups and Space Groups
Crystallographic Point Groups and Space Groups Physics 251 Spring 2011 Matt Wittmann University of California Santa Cruz June 8, 2011 Mathematical description of a crystal Definition A Bravais lattice
More informationBrownian Motion and Langevin Equations
1 Brownian Motion and Langevin Equations 1.1 Langevin Equation and the Fluctuation- Dissipation Theorem The theory of Brownian motion is perhaps the simplest approximate way to treat the dynamics of nonequilibrium
More information3. General properties of phase transitions and the Landau theory
3. General properties of phase transitions and the Landau theory In this Section we review the general properties and the terminology used to characterise phase transitions, which you will have already
More informationarxiv:cond-mat/ v2 7 Dec 1999
Molecular dynamics study of a classical two-dimensional electron system: Positional and orientational orders arxiv:cond-mat/9906213v2 7 Dec 1999 Satoru Muto 1, Hideo Aoki Department of Physics, University
More informationPhysics PhD Qualifying Examination Part I Wednesday, January 21, 2015
Physics PhD Qualifying Examination Part I Wednesday, January 21, 2015 Name: (please print) Identification Number: STUDENT: Designate the problem numbers that you are handing in for grading in the appropriate
More informationGlasses and Jamming: lessons from hard disks in a narrow channel
Glasses and Jamming: lessons from hard disks in a narrow channel Mike Moore School of Physics and Astronomy, University of Manchester, UK June 2014 Co-Author: Mike Godfrey, University of Manchester Mike
More informationPart II Statistical Physics
Part II Statistical Physics Theorems Based on lectures by H. S. Reall Notes taken by Dexter Chua Lent 2017 These notes are not endorsed by the lecturers, and I have modified them (often significantly)
More informationGibb s free energy change with temperature in a single component system
Gibb s free energy change with temperature in a single component system An isolated system always tries to maximize the entropy. That means the system is stable when it has maximum possible entropy. Instead
More informationTemperature Fluctuations and Entropy Formulas
Temperature Fluctuations and Entropy Formulas Does it or does not? T.S. Biró G.G. Barnaföldi P. Ván MTA Heavy Ion Research Group Research Centre for Physics, Budapest February 1, 2014 J.Uffink, J.van Lith:
More informationThe goal of equilibrium statistical mechanics is to calculate the diagonal elements of ˆρ eq so we can evaluate average observables < A >= Tr{Â ˆρ eq
Chapter. The microcanonical ensemble The goal of equilibrium statistical mechanics is to calculate the diagonal elements of ˆρ eq so we can evaluate average observables < A >= Tr{Â ˆρ eq } = A that give
More informationExperimental Soft Matter (M. Durand, G. Foffi)
Master 2 PCS/PTSC 2016-2017 10/01/2017 Experimental Soft Matter (M. Durand, G. Foffi) Nota Bene Exam duration : 3H ecture notes are not allowed. Electronic devices (including cell phones) are prohibited,
More informationMagnetic ordering of local moments
Magnetic ordering Types of magnetic structure Ground state of the Heisenberg ferromagnet and antiferromagnet Spin wave High temperature susceptibility Mean field theory Magnetic ordering of local moments
More informationRare Event Simulations
Rare Event Simulations Homogeneous nucleation is a rare event (e.g. Liquid Solid) Crystallization requires supercooling (µ solid < µ liquid ) Crystal nucleus 2r Free - energy gain r 3 4! 3! GBulk = ñr!
More informationLine adsorption in a mean-field density functional model
Chemistry Physical & Theoretical Chemistry fields Okayama University Year 2006 Line adsorption in a mean-field density functional model Kenichiro Koga Benjamin Widom Okayama University, koga@cc.okayama-u.ac.jp
More informationINTRODUCTION TO о JLXJLA Из А lv-/xvj_y JrJrl Y üv_>l3 Second Edition
INTRODUCTION TO о JLXJLA Из А lv-/xvj_y JrJrl Y üv_>l3 Second Edition Kerson Huang CRC Press Taylor & Francis Croup Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group an Informa
More information