Boring (?) first-order phase transitions

Size: px
Start display at page:

Download "Boring (?) first-order phase transitions"

Transcription

1 Boring (?) first-order phase transitions Des Johnston Edinburgh, June 2014 Johnston First Order 1/34

2 Plan of talk First and Second Order Transitions Finite size scaling (FSS) at first order transitions Non-standard FSS Johnston First Order 2/34

3 First and Second Order Transitions (Ehrenfest) First-order phase transitions exhibit a discontinuity in the first derivative of the free energy with respect to some thermodynamic variable. Second-order transitions are continuous in the first derivative but exhibit discontinuity in a second derivative of the free energy. Johnston First Order 3/34

4 First and Second Order Transitions ( modern ) First-order phase transitions are those that involve a latent heat. Second-order transitions are also called continuous phase transitions. They are characterized by a divergent susceptibility, an infinite correlation length, and a power-law decay of correlations near criticality. Johnston First Order 4/34

5 First Order Transitions - Piccies First order - melting First order - field driven Johnston First Order 5/34

6 Second Order Transition H = ij σ i σ j ; Z(β) = {σ} exp( βh) Johnston First Order 6/34

7 First and Second Order Transitions - Piccies First order - discontinuities in magnetization, energy (latent heat) Second order - divergences in specific heat, susceptibility Johnston First Order 7/34

8 Looking at transitions Cook up a (lattice) model Identify order parameter, measure/calculate things Look for transition Continuous - extract critical exponents Finite Size Scaling (FSS) Johnston First Order 8/34

9 Define a model: q-state Potts Hamiltonian H q = ij δ σi,σ j Ising : H = σ i σ j ij Evaluate a partition function Z(β) = {σ} exp( βh q ) Derivatives of free energy give observables (energy, magnetization..) F (β) = ln Z(β) Johnston First Order 9/34

10 Measure 1001 Different Observables Order parameter M = (q max{n i } N)/(q 1) Per-site quantities denoted by e = E/N and m = M/N u(β) = E /N, C(β) = β 2 N[ e 2 e 2 ], B(β) = [1 e4 3 e 2 2 ]. m(β) = m, χ(β) = β N[ m 2 m 2 ], U 2 (β) = [1 m2 3 m 2 ]. Johnston First Order 10/34

11 Continuous Transitions - Critical exponents (Continuous) Phase transitions characterized by critical exponents Define t = T T c /T c Then in general, ξ t ν, M t β, C t α, χ t γ Can be rephrased in terms of the linear size of a system L or N 1/d ξ N 1/d, M N β/νd, C N α/νd, χ N γ/νd Johnston First Order 11/34

12 Scaling Another exponent... ψ(0)ψ(r) r d+2 η Scaling relations α = 2 νd ; α + 2β + γ = 2 Two independent exponents Johnston First Order 12/34

13 Now, first order... Formally νd = 1, α = 1 i.e. Volume scaling C N ; ξ N Squeezing a delta function onto a lattice Johnston First Order 13/34

14 What does a first order system look like (at PT) I? Johnston First Order 14/34

15 What does a first order system look like (at PT) II? E Hysteresis COLD HOT β Johnston First Order 15/34

16 What does a first order system look like (at PT) III? Phase coexistence P(E) e-05 1e-06 1e E Johnston First Order 16/34

17 Heuristic two-phase model A fraction W o in q ordered phase(s), energy ê o A fraction W d = 1 W o in disordered phase, energy ê d The hat = quantities evaluated at β Neglect fluctuations within the phases Johnston First Order 17/34

18 Energy moments Energy moments become e n = W o ê n o + (1 W o )ê n d And the specific heat then reads: C V (β, L) = L d β 2 ( e 2 e 2) = L d β 2 W o (1 W o ) ê 2 Max of C max V = L d (β ê/2) 2 at W o = W d = 0.5 Volume scaling Johnston First Order 18/34

19 FSS: Specific Heat Probability of being in any of the states p o e βld ˆf o and p d e βld ˆf d Time spent in the ordered states qp o Expand around β Solve for specific heat peak W o /W d qe Ld β ˆf o /e βld ˆf d 0 = ln q + L d ê(β β ) +... β Cmax V (L) = β ln q L d ê +... Johnston First Order 19/34

20 FSS: Binder Cumulant Energetic Binder cumulant Use (again) B(β, L) = 1 e4 3 e 2 2 e n = W o ê n o + (1 W o )ê n d to get location of min: β Bmin (L) β Bmin (L) = β ln(qê2 o/ê 2 d ) L d ê L d FSS Johnston First Order 20/34

21 An Aside You can do this more carefully Pirogov-Sinai Theory (Borgs/Kotecký) Z(β) = ] [ ] [e βld f d + qe βld f o 1 + O(L d e L/L 0 ) Z(β) 2 ( qe βld (f d +f 0 βl )/2 d (f d f 0 ) cosh + 1 ) 2 2 ln q x = βld (f d f 0 ) ln q Ld ê(β β ) ln q +... Johnston First Order 21/34

22 A Bit Dull... Peaks grow like L d = V = N Critical temperatures shift like 1/L d No other numbers (?) Johnston First Order 22/34

23 Some other numbers - Fixed BC Fixed boundary conditions Z(β) = [ e β(ld f d +2dL d 1 f d ) + qe β(ld f o+2dl d 1 f o) ] [1 +...] x = Ld (f d f 0 ) + dl d 1 ( 2 f d f 0 ) ln q al d (β β ) + bl d (since f d = a 1 + ẽ d (β β ) +..., fo = a 2 + ẽ o (β β ) +...) β 1 L It is clear that there can now be O(1/L) corrections Johnston First Order 23/34

24 Fitting the data - 8-state Potts Estimated from peak in χ: β c (L) = β + a L +... Johnston First Order 24/34

25 Some other numbers - degeneracy A 3D plaquette Ising model Its dual A big critical temperature discrepancy (30σ) Johnston First Order 25/34

26 A 3D Plaquette Ising action 3D cubic, spins on vertices H = 1 σ i σ j σ k σ l 2 [i,j,k,l] NOT H = U ij U jk U kl U li, U ij = ±1 Z 2 Lattice Gauge [i,j,k,l] Johnston First Order 26/34

27 And the dual An anisotropically coupled Ashkin-Teller model H dual = 1 σ i σ j 1 τ i τ j 1 σ i σ j τ i τ j, ij x ij y ij z Johnston First Order 27/34

28 The Problem Original model: L = 8, 9,..., 26, 27, periodic bc, 1/L d fits β = (30) Dual model: L = 8, 10,..., 22, 24, periodic bc, 1/L d fits βdual = (19) β = (11) Estimates are about 30 error bars apart. Johnston First Order 28/34

29 A Solution... Degeneracy Modified FSS Johnston First Order 29/34

30 Groundstates: Plaquette Persists into low temperature phase: degeneracy 2 3L Johnston First Order 30/34

31 Ground state Johnston First Order 31/34

32 1st Order FSS with Exponential Degeneracy Normally q is constant Suppose instead q e L β Cmax V (L) = β ln q L d ê +... become β Bmin (L) = β ln(qê2 o/ê 2 d ) L d ê +... β Cmax V (L) = β 1 L d 1 ê +... β Bmin (L) = β ln(ê2 o/ê 2 d ) L d 1 ê +... Johnston First Order 32/34

33 Conclusions Standard 1st order FSS: 1/L 3 corrections in 3D Fixed BC: 1/L (surface tension) Exponential degeneracy: 1/L 2 in 3D Further applications may be higher-dimensional variants of the gonihedric model, ANNNI models, spin ice systems, orbital compass models,... Johnston First Order 33/34

34 References K. Binder, Rep. Prog. Phys. 50, 783 (1987) C. Borgs and R. Kotecký, Phys. Rev. Lett. 68, 1734 (1992) W. Janke, Phys. Rev. B 47, (1993) M. Mueller, W. Janke and D. A. Johnston, Phys. Rev. Lett. 112, (2014) Johnston First Order 34/34

Disorder-induced rounding of the phase transition. in the large-q-state Potts model. F. Iglói SZFKI - Budapest

Disorder-induced rounding of the phase transition. in the large-q-state Potts model. F. Iglói SZFKI - Budapest Disorder-induced rounding of the phase transition in the large-q-state Potts model M.T. Mercaldo J-C. Anglès d Auriac Università di Salerno CNRS - Grenoble F. Iglói SZFKI - Budapest Motivations 2. CRITICAL

More information

Thermal transport in the anisotropic Heisenberg chain with S = 1/2 and nearest-neighbor interactions

Thermal transport in the anisotropic Heisenberg chain with S = 1/2 and nearest-neighbor interactions Thermal transport in the anisotropic Heisenberg chain with S = 1/2 and nearest-neighbor interactions D. L. Huber Department of Physics, University of Wisconsin-Madison, Madison, WI 53706 Abstract The purpose

More information

arxiv:hep-lat/9210041v1 30 Oct 1992

arxiv:hep-lat/9210041v1 30 Oct 1992 1 The Interface Tension in Quenched QCD at the Critical Temperature B. Grossmann a, M.. aursen a, T. Trappenberg a b and U. J. Wiese c a HRZ, c/o Kfa Juelich, P.O. Box 1913, D-5170 Jülich, Germany arxiv:hep-lat/9210041v1

More information

( ) = ( ) = {,,, } β ( ), < 1 ( ) + ( ) = ( ) + ( )

( ) = ( ) = {,,, } β ( ), < 1 ( ) + ( ) = ( ) + ( ) { } ( ) = ( ) = {,,, } ( ) β ( ), < 1 ( ) + ( ) = ( ) + ( ) max, ( ) [ ( )] + ( ) [ ( )], [ ( )] [ ( )] = =, ( ) = ( ) = 0 ( ) = ( ) ( ) ( ) =, ( ), ( ) =, ( ), ( ). ln ( ) = ln ( ). + 1 ( ) = ( ) Ω[ (

More information

Complex dynamics made simple: colloidal dynamics

Complex dynamics made simple: colloidal dynamics Complex dynamics made simple: colloidal dynamics Chemnitz, June 2010 Complex dynamics made simple: colloidal dynamics Chemnitz, June 2010 Terminology Quench: sudden change of external parameter, e.g.

More information

Frustrated magnetism on Hollandite lattice

Frustrated magnetism on Hollandite lattice Frustrated magnetism on Hollandite lattice Saptarshi Mandal (ICTP, Trieste, Italy) Acknowledgment: A. Andreanov(MPIKS, Dresden) Y. Crespo and N. Seriani(ICTP, Italy) Workshop on Current Trends in Frustrated

More information

The Quantum Harmonic Oscillator Stephen Webb

The Quantum Harmonic Oscillator Stephen Webb The Quantum Harmonic Oscillator Stephen Webb The Importance of the Harmonic Oscillator The quantum harmonic oscillator holds a unique importance in quantum mechanics, as it is both one of the few problems

More information

Critical Phenomena and Percolation Theory: I

Critical Phenomena and Percolation Theory: I Critical Phenomena and Percolation Theory: I Kim Christensen Complexity & Networks Group Imperial College London Joint CRM-Imperial College School and Workshop Complex Systems Barcelona 8-13 April 2013

More information

Euclidean quantum gravity revisited

Euclidean quantum gravity revisited Institute for Gravitation and the Cosmos, Pennsylvania State University 15 June 2009 Eastern Gravity Meeting, Rochester Institute of Technology Based on: First-order action and Euclidean quantum gravity,

More information

Section 3: Crystal Binding

Section 3: Crystal Binding Physics 97 Interatomic forces Section 3: rystal Binding Solids are stable structures, and therefore there exist interactions holding atoms in a crystal together. For example a crystal of sodium chloride

More information

PUBLIC TRANSPORT SYSTEMS IN POLAND: FROM BIAŁYSTOK TO ZIELONA GÓRA BY BUS AND TRAM USING UNIVERSAL STATISTICS OF COMPLEX NETWORKS

PUBLIC TRANSPORT SYSTEMS IN POLAND: FROM BIAŁYSTOK TO ZIELONA GÓRA BY BUS AND TRAM USING UNIVERSAL STATISTICS OF COMPLEX NETWORKS Vol. 36 (2005) ACTA PHYSICA POLONICA B No 5 PUBLIC TRANSPORT SYSTEMS IN POLAND: FROM BIAŁYSTOK TO ZIELONA GÓRA BY BUS AND TRAM USING UNIVERSAL STATISTICS OF COMPLEX NETWORKS Julian Sienkiewicz and Janusz

More information

Bond-correlated percolation model and the unusual behaviour of supercooled water

Bond-correlated percolation model and the unusual behaviour of supercooled water J. Phys. A: Math. Gen. 16 (1983) L321-L326. Printed in Great Britain LETTER TO THE EDITOR Bond-correlated percolation model and the unusual behaviour of supercooled water Chin-Kun Hu Lash-Miller Chemical

More information

NMR and IR spectra & vibrational analysis

NMR and IR spectra & vibrational analysis Lab 5: NMR and IR spectra & vibrational analysis A brief theoretical background 1 Some of the available chemical quantum methods for calculating NMR chemical shifts are based on the Hartree-Fock self-consistent

More information

The Power (Law) of Indian Markets: Analysing NSE and BSE Trading Statistics

The Power (Law) of Indian Markets: Analysing NSE and BSE Trading Statistics The Power (Law) of Indian Markets: Analysing NSE and BSE Trading Statistics Sitabhra Sinha and Raj Kumar Pan The Institute of Mathematical Sciences, C. I. T. Campus, Taramani, Chennai - 6 113, India. sitabhra@imsc.res.in

More information

Hot billet charging saves millions of energy costs and helps to increase rolling mill productivity considerably.

Hot billet charging saves millions of energy costs and helps to increase rolling mill productivity considerably. Hot billet charging saves millions of energy costs and helps to increase rolling mill productivity considerably. Authors: Volker Preiß - Billet Yard Operation Manager BSW Sebastian Ritter Project Engineer

More information

GPU Accelerated Monte Carlo Simulations and Time Series Analysis

GPU Accelerated Monte Carlo Simulations and Time Series Analysis GPU Accelerated Monte Carlo Simulations and Time Series Analysis Institute of Physics, Johannes Gutenberg-University of Mainz Center for Polymer Studies, Department of Physics, Boston University Artemis

More information

Small window overlaps are effective probes of replica symmetry breaking in three-dimensional spin glasses

Small window overlaps are effective probes of replica symmetry breaking in three-dimensional spin glasses J. Phys. A: Math. Gen. 31 (1998) L481 L487. Printed in the UK PII: S0305-4470(98)93259-0 LETTER TO THE EDITOR Small window overlaps are effective probes of replica symmetry breaking in three-dimensional

More information

Statistical Physics Exam

Statistical Physics Exam Statistical Physics Exam 23rd April 24 Name Student Number Problem Problem 2 Problem 3 Problem 4 Total Percentage Mark Useful constants gas constant R Boltzmann constant k B Avogadro number N A speed of

More information

ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE

ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE YUAN TIAN This synopsis is designed merely for keep a record of the materials covered in lectures. Please refer to your own lecture notes for all proofs.

More information

DOMAIN WALLS, SURFACE TENSION AND WELTING IN THE THREE-DIMENSIONAL THREE-STATE POTTS MODEL

DOMAIN WALLS, SURFACE TENSION AND WELTING IN THE THREE-DIMENSIONAL THREE-STATE POTTS MODEL Nuclear Physics B350 (1991) 563-588 North-Holland DOMAIN WALLS, SURFACE TENSION AND WELTING IN THE THREE-DIMENSIONAL THREE-STATE POTTS MODEL F. KARSCH AND A. PATKOS* CERN, Theory Dirision, 1211 Genet-a

More information

Sucesses and limitations of dynamical mean field theory. A.-M. Tremblay G. Sordi, D. Sénéchal, K. Haule, S. Okamoto, B. Kyung, M.

Sucesses and limitations of dynamical mean field theory. A.-M. Tremblay G. Sordi, D. Sénéchal, K. Haule, S. Okamoto, B. Kyung, M. Sucesses and limitations of dynamical mean field theory A.-M. Tremblay G. Sordi, D. Sénéchal, K. Haule, S. Okamoto, B. Kyung, M. Civelli MIT, 20 October, 2011 How to make a metal Courtesy, S. Julian r

More information

The first law: transformation of energy into heat and work. Chemical reactions can be used to provide heat and for doing work.

The first law: transformation of energy into heat and work. Chemical reactions can be used to provide heat and for doing work. The first law: transformation of energy into heat and work Chemical reactions can be used to provide heat and for doing work. Compare fuel value of different compounds. What drives these reactions to proceed

More information

MIT 2.810 Manufacturing Processes and Systems. Homework 6 Solutions. Casting. October 15, 2015. Figure 1: Casting defects

MIT 2.810 Manufacturing Processes and Systems. Homework 6 Solutions. Casting. October 15, 2015. Figure 1: Casting defects MIT 2.810 Manufacturing Processes and Systems Casting October 15, 2015 Problem 1. Casting defects. (a) Figure 1 shows various defects and discontinuities in cast products. Review each one and offer solutions

More information

No Evidence for a new phase of dense hydrogen above 325 GPa

No Evidence for a new phase of dense hydrogen above 325 GPa 1 No Evidence for a new phase of dense hydrogen above 325 GPa Ranga P. Dias, Ori Noked, and Isaac F. Silvera Lyman Laboratory of Physics, Harvard University, Cambridge MA, 02138 In recent years there has

More information

Solving Exponential Equations

Solving Exponential Equations Solving Exponential Equations Deciding How to Solve Exponential Equations When asked to solve an exponential equation such as x + 6 = or x = 18, the first thing we need to do is to decide which way is

More information

FIELD THEORY OF ISING PERCOLATING CLUSTERS

FIELD THEORY OF ISING PERCOLATING CLUSTERS UK Meeting on Integrable Models and Conformal Field heory University of Kent, Canterbury 16-17 April 21 FIELD HEORY OF ISING PERCOLAING CLUSERS Gesualdo Delfino SISSA-rieste Based on : GD, Nucl.Phys.B

More information

1 Introduction. 2 Prediction with Expert Advice. Online Learning 9.520 Lecture 09

1 Introduction. 2 Prediction with Expert Advice. Online Learning 9.520 Lecture 09 1 Introduction Most of the course is concerned with the batch learning problem. In this lecture, however, we look at a different model, called online. Let us first compare and contrast the two. In batch

More information

arxiv:quant-ph/0211152v1 23 Nov 2002

arxiv:quant-ph/0211152v1 23 Nov 2002 arxiv:quant-ph/0211152v1 23 Nov 2002 SCALABLE ARCHITECTURE FOR ADIABATIC QUANTUM COMPUTING OF NP-HARD PROBLEMS William M. Kaminsky* and Seth Lloyd Massachusetts Institute of Technology, Cambridge, MA 02139

More information

arxiv:cond-mat/9809050v1 [cond-mat.stat-mech] 2 Sep 1998

arxiv:cond-mat/9809050v1 [cond-mat.stat-mech] 2 Sep 1998 arxiv:cond-mat/9809050v1 [cond-mat.stat-mech] 2 Sep 1998 One-dimensional Ising model with long-range and random short-range interactions A. P. Vieira and L. L. Gonçalves Departamento de Física da UFC,

More information

Modelling Emergence of Money

Modelling Emergence of Money Vol. 117 (2010) ACTA PHYSICA POLONICA A No. 4 Proceedings of the 4th Polish Symposium on Econo- and Sociophysics, Rzeszów, Poland, May 7 9, 2009 Modelling Emergence of Money A.Z. Górski a, S. Drożdż a,b

More information

Methods of Data Analysis Working with probability distributions

Methods of Data Analysis Working with probability distributions Methods of Data Analysis Working with probability distributions Week 4 1 Motivation One of the key problems in non-parametric data analysis is to create a good model of a generating probability distribution,

More information

Claudio J. Tessone. Pau Amengual. Maxi San Miguel. Raúl Toral. Horacio Wio. Eur. Phys. J. B 39, 535 (2004) http://www.imedea.uib.

Claudio J. Tessone. Pau Amengual. Maxi San Miguel. Raúl Toral. Horacio Wio. Eur. Phys. J. B 39, 535 (2004) http://www.imedea.uib. Horacio Wio Raúl Toral Eur. Phys. J. B 39, 535 (2004) Claudio J. Tessone Pau Amengual Maxi San Miguel http://www.imedea.uib.es/physdept Models of Consensus vs. Polarization, or Segregation: Voter model,

More information

Review of Statistical Mechanics

Review of Statistical Mechanics Review of Statistical Mechanics 3. Microcanonical, Canonical, Grand Canonical Ensembles In statistical mechanics, we deal with a situation in which even the quantum state of the system is unknown. The

More information

PHYS 1624 University Physics I. PHYS 2644 University Physics II

PHYS 1624 University Physics I. PHYS 2644 University Physics II PHYS 1624 Physics I An introduction to mechanics, heat, and wave motion. This is a calculus- based course for Scientists and Engineers. 4 hours (3 lecture/3 lab) Prerequisites: Credit for MATH 2413 (Calculus

More information

Implications of an inverse branching aftershock sequence model

Implications of an inverse branching aftershock sequence model Implications of an inverse branching aftershock sequence model D. L. Turcotte* and S. G. Abaimov Department of Geology, University of California, Davis, California 95616, USA I. Dobson Electrical and Computer

More information

The Network Structure of Hard Combinatorial Landscapes

The Network Structure of Hard Combinatorial Landscapes The Network Structure of Hard Combinatorial Landscapes Marco Tomassini 1, Sebastien Verel 2, Gabriela Ochoa 3 1 University of Lausanne, Lausanne, Switzerland 2 University of Nice Sophia-Antipolis, France

More information

8. Time Series and Prediction

8. Time Series and Prediction 8. Time Series and Prediction Definition: A time series is given by a sequence of the values of a variable observed at sequential points in time. e.g. daily maximum temperature, end of day share prices,

More information

N 1. (q k+1 q k ) 2 + α 3. k=0

N 1. (q k+1 q k ) 2 + α 3. k=0 Teoretisk Fysik Hand-in problem B, SI1142, Spring 2010 In 1955 Fermi, Pasta and Ulam 1 numerically studied a simple model for a one dimensional chain of non-linear oscillators to see how the energy distribution

More information

MBA Jump Start Program

MBA Jump Start Program MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Online Appendix: Basic Mathematical Concepts 2 1 The Number Spectrum Generally we depict numbers increasing from left to right

More information

Thermodynamics is the study of heat. It s what comes into play when you drop an ice cube

Thermodynamics is the study of heat. It s what comes into play when you drop an ice cube Chapter 12 You re Getting Warm: Thermodynamics In This Chapter Converting between temperature scales Working with linear expansion Calculating volume expansion Using heat capacities Understanding latent

More information

Orbital Dynamics coupled with Jahn-Teller phonons in Strongly Correlated Electron System

Orbital Dynamics coupled with Jahn-Teller phonons in Strongly Correlated Electron System The 5 th Scienceweb GCOE International Symposium 1 Orbital Dynamics coupled with Jahn-Teller phonons in Strongly Correlated Electron System Department of Physics, Tohoku University Joji Nasu In collaboration

More information

Lecture 36 (Walker 18.8,18.5-6,)

Lecture 36 (Walker 18.8,18.5-6,) Lecture 36 (Walker 18.8,18.5-6,) Entropy 2 nd Law of Thermodynamics Dec. 11, 2009 Help Session: Today, 3:10-4:00, TH230 Review Session: Monday, 3:10-4:00, TH230 Solutions to practice Lecture 36 final on

More information

4. Thermodynamics of Polymer Blends

4. Thermodynamics of Polymer Blends 4. Thermodynamics of Polymer Blends Polymeric materials find growing applications in various fields of everyday life because they offer a wide range of application relevant properties. Blending of polymers

More information

Supplementary Notes on Entropy and the Second Law of Thermodynamics

Supplementary Notes on Entropy and the Second Law of Thermodynamics ME 4- hermodynamics I Supplementary Notes on Entropy and the Second aw of hermodynamics Reversible Process A reversible process is one which, having taken place, can be reversed without leaving a change

More information

New parameterization of cloud optical properties

New parameterization of cloud optical properties New parameterization of cloud optical properties proposed for model ALARO-0 Results of Prague LACE stay 1.8. 1.1005 under scientific supervision of Jean-François Geleyn J. Mašek, 11.1005 Main target of

More information

A box-covering algorithm for fractal scaling in scale-free networks

A box-covering algorithm for fractal scaling in scale-free networks CHAOS 17, 026116 2007 A box-covering algorithm for fractal scaling in scale-free networks J. S. Kim CTP & FPRD, School of Physics and Astronomy, Seoul National University, NS50, Seoul 151-747, Korea K.-I.

More information

Tobias Märkl. November 16, 2009

Tobias Märkl. November 16, 2009 ,, Tobias Märkl to 1/f November 16, 2009 1 / 33 Content 1 duction to of Statistical Comparison to Other Types of Noise of of 2 Random duction to Random General of, to 1/f 3 4 2 / 33 , to 1/f 3 / 33 What

More information

G U I D E T O A P P L I E D O R B I T A L M E C H A N I C S F O R K E R B A L S P A C E P R O G R A M

G U I D E T O A P P L I E D O R B I T A L M E C H A N I C S F O R K E R B A L S P A C E P R O G R A M G U I D E T O A P P L I E D O R B I T A L M E C H A N I C S F O R K E R B A L S P A C E P R O G R A M CONTENTS Foreword... 2 Forces... 3 Circular Orbits... 8 Energy... 10 Angular Momentum... 13 FOREWORD

More information

Heat equation examples

Heat equation examples Heat equation examples The Heat equation is discussed in depth in http://tutorial.math.lamar.edu/classes/de/intropde.aspx, starting on page 6. You may recall Newton s Law of Cooling from Calculus. Just

More information

Physics 5D - Nov 18, 2013

Physics 5D - Nov 18, 2013 Physics 5D - Nov 18, 2013 30 Midterm Scores B } Number of Scores 25 20 15 10 5 F D C } A- A A + 0 0-59.9 60-64.9 65-69.9 70-74.9 75-79.9 80-84.9 Percent Range (%) The two problems with the fewest correct

More information

Physics and Economy of Energy Storage

Physics and Economy of Energy Storage International Conference Energy Autonomy through Storage of Renewable Energies by EUROSOLAR and WCRE October 30 and 31, 2006 Gelsenkirchen / Germany Physics and Economy of Energy Storage Ulf Bossel European

More information

L5. P1. Lecture 5. Solids. The free electron gas

L5. P1. Lecture 5. Solids. The free electron gas Lecture 5 Page 1 Lecture 5 L5. P1 Solids The free electron gas In a solid state, a few loosely bound valence (outermost and not in completely filled shells) elections become detached from atoms and move

More information

The atomic packing factor is defined as the ratio of sphere volume to the total unit cell volume, or APF = V S V C. = 2(sphere volume) = 2 = V C = 4R

The atomic packing factor is defined as the ratio of sphere volume to the total unit cell volume, or APF = V S V C. = 2(sphere volume) = 2 = V C = 4R 3.5 Show that the atomic packing factor for BCC is 0.68. The atomic packing factor is defined as the ratio of sphere volume to the total unit cell volume, or APF = V S V C Since there are two spheres associated

More information

Statistical properties of trading activity in Chinese Stock Market

Statistical properties of trading activity in Chinese Stock Market Physics Procedia 3 (2010) 1699 1706 Physics Procedia 00 (2010) 1 8 Physics Procedia www.elsevier.com/locate/procedia Statistical properties of trading activity in Chinese Stock Market Xiaoqian Sun a, Xueqi

More information

THE STATISTICAL TREATMENT OF EXPERIMENTAL DATA 1

THE STATISTICAL TREATMENT OF EXPERIMENTAL DATA 1 THE STATISTICAL TREATMET OF EXPERIMETAL DATA Introduction The subject of statistical data analysis is regarded as crucial by most scientists, since error-free measurement is impossible in virtually all

More information

Université Lille 1 Sciences et Technologies, Lille, France Lille Laboratoire de Physique des Lasers, Atomes et Molécules Équipe Chaos Quantique

Université Lille 1 Sciences et Technologies, Lille, France Lille Laboratoire de Physique des Lasers, Atomes et Molécules Équipe Chaos Quantique Université Lille 1 Sciences et Technologies, Lille, France Lille Laboratoire de Physique des Lasers, Atomes et Molécules Équipe Chaos Quantique FRISNO 11 Aussois 1/4/011 Quantum simulators: The Anderson

More information

Physics 2326 - Assignment No: 3 Chapter 22 Heat Engines, Entropy and Second Law of Thermodynamics

Physics 2326 - Assignment No: 3 Chapter 22 Heat Engines, Entropy and Second Law of Thermodynamics Serway/Jewett: PSE 8e Problems Set Ch. 22-1 Physics 2326 - Assignment No: 3 Chapter 22 Heat Engines, Entropy and Second Law of Thermodynamics Objective Questions 1. A steam turbine operates at a boiler

More information

Fundamentals of grain boundaries and grain boundary migration

Fundamentals of grain boundaries and grain boundary migration 1. Fundamentals of grain boundaries and grain boundary migration 1.1. Introduction The properties of crystalline metallic materials are determined by their deviation from a perfect crystal lattice, which

More information

CryoEDM A Cryogenic Neutron-EDM Experiment. Collaboration: Sussex University, RAL, ILL, Kure University, Oxford University Hans Kraus

CryoEDM A Cryogenic Neutron-EDM Experiment. Collaboration: Sussex University, RAL, ILL, Kure University, Oxford University Hans Kraus CryoEDM A Cryogenic Neutron-EDM Experiment Collaboration: Sussex University, RAL, ILL, Kure University, Oxford University Hans Kraus nedm Overview Theoretical Background The Method of Ramsey Resonance

More information

6 J - vector electric current density (A/m2 )

6 J - vector electric current density (A/m2 ) Determination of Antenna Radiation Fields Using Potential Functions Sources of Antenna Radiation Fields 6 J - vector electric current density (A/m2 ) M - vector magnetic current density (V/m 2 ) Some problems

More information

Quantitative Analysis of Foreign Exchange Rates

Quantitative Analysis of Foreign Exchange Rates Quantitative Analysis of Foreign Exchange Rates Alexander Becker, Ching-Hao Wang Boston University, Department of Physics (Dated: today) In our class project we have explored foreign exchange data. We

More information

Quantum Monte Carlo and the negative sign problem

Quantum Monte Carlo and the negative sign problem Quantum Monte Carlo and the negative sign problem or how to earn one million dollar Matthias Troyer, ETH Zürich Uwe-Jens Wiese, Universität Bern Complexity of many particle problems Classical 1 particle:

More information

Detrending Moving Average Algorithm: from finance to genom. materials

Detrending Moving Average Algorithm: from finance to genom. materials Detrending Moving Average Algorithm: from finance to genomics and disordered materials Anna Carbone Physics Department Politecnico di Torino www.polito.it/noiselab September 27, 2009 Publications 1. Second-order

More information

Simulation of Spin Models on Nvidia Graphics Cards using CUDA

Simulation of Spin Models on Nvidia Graphics Cards using CUDA Simulation of Spin Models on Nvidia Graphics Cards using CUDA D I P L O M A R B E I T zur Erlangung des akademischen Grades Diplom-Physiker (Dipl.-Phys.) im Fach Physik eingereicht an der Mathematisch-Naturwissenschaftlichen

More information

Understanding Poles and Zeros

Understanding Poles and Zeros MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.14 Analysis and Design of Feedback Control Systems Understanding Poles and Zeros 1 System Poles and Zeros The transfer function

More information

We will study the temperature-pressure diagram of nitrogen, in particular the triple point.

We will study the temperature-pressure diagram of nitrogen, in particular the triple point. K4. Triple Point of Nitrogen I. OBJECTIVE OF THE EXPERIMENT We will study the temperature-pressure diagram of nitrogen, in particular the triple point. II. BAKGROUND THOERY States of matter Matter is made

More information

Measurement and Simulation of Electron Thermal Transport in the MST Reversed-Field Pinch

Measurement and Simulation of Electron Thermal Transport in the MST Reversed-Field Pinch 1 EX/P3-17 Measurement and Simulation of Electron Thermal Transport in the MST Reversed-Field Pinch D. J. Den Hartog 1,2, J. A. Reusch 1, J. K. Anderson 1, F. Ebrahimi 1,2,*, C. B. Forest 1,2 D. D. Schnack

More information

Productioin OVERVIEW. WSG5 7/7/03 4:35 PM Page 63. Copyright 2003 by Academic Press. All rights of reproduction in any form reserved.

Productioin OVERVIEW. WSG5 7/7/03 4:35 PM Page 63. Copyright 2003 by Academic Press. All rights of reproduction in any form reserved. WSG5 7/7/03 4:35 PM Page 63 5 Productioin OVERVIEW This chapter reviews the general problem of transforming productive resources in goods and services for sale in the market. A production function is the

More information

Second Law of Thermodynamics Alternative Statements

Second Law of Thermodynamics Alternative Statements Second Law of Thermodynamics Alternative Statements There is no simple statement that captures all aspects of the second law. Several alternative formulations of the second law are found in the technical

More information

Matrix Algebra and Applications

Matrix Algebra and Applications Matrix Algebra and Applications Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 1 / 49 EC2040 Topic 2 - Matrices and Matrix Algebra Reading 1 Chapters

More information

Chemistry 433. The Third Law of Thermodynamics. Residual Entropy. CO: an Imperfect Crystal. Question. Question. Lecture 12 The Third Law

Chemistry 433. The Third Law of Thermodynamics. Residual Entropy. CO: an Imperfect Crystal. Question. Question. Lecture 12 The Third Law Chemistry 433 Lecture 12 he hird Law he hird Law of hermodynamics he third law of thermodynamics states that every substance has a positive entropy, but at zero Kelvin the entropy is zero for a perfectly

More information

5.1 Simple and Compound Interest

5.1 Simple and Compound Interest 5.1 Simple and Compound Interest Question 1: What is simple interest? Question 2: What is compound interest? Question 3: What is an effective interest rate? Question 4: What is continuous compound interest?

More information

NMR Measurement of T1-T2 Spectra with Partial Measurements using Compressive Sensing

NMR Measurement of T1-T2 Spectra with Partial Measurements using Compressive Sensing NMR Measurement of T1-T2 Spectra with Partial Measurements using Compressive Sensing Alex Cloninger Norbert Wiener Center Department of Mathematics University of Maryland, College Park http://www.norbertwiener.umd.edu

More information

NONLINEAR TIME SERIES ANALYSIS

NONLINEAR TIME SERIES ANALYSIS NONLINEAR TIME SERIES ANALYSIS HOLGER KANTZ AND THOMAS SCHREIBER Max Planck Institute for the Physics of Complex Sy stems, Dresden I CAMBRIDGE UNIVERSITY PRESS Preface to the first edition pug e xi Preface

More information

Chapter 10 Liquids & Solids

Chapter 10 Liquids & Solids 1 Chapter 10 Liquids & Solids * 10.1 Polar Covalent Bonds & Dipole Moments - van der Waals constant for water (a = 5.28 L 2 atm/mol 2 ) vs O 2 (a = 1.36 L 2 atm/mol 2 ) -- water is polar (draw diagram)

More information

GCMs with Implicit and Explicit cloudrain processes for simulation of extreme precipitation frequency

GCMs with Implicit and Explicit cloudrain processes for simulation of extreme precipitation frequency GCMs with Implicit and Explicit cloudrain processes for simulation of extreme precipitation frequency In Sik Kang Seoul National University Young Min Yang (UH) and Wei Kuo Tao (GSFC) Content 1. Conventional

More information

Free Electron Fermi Gas (Kittel Ch. 6)

Free Electron Fermi Gas (Kittel Ch. 6) Free Electron Fermi Gas (Kittel Ch. 6) Role of Electrons in Solids Electrons are responsible for binding of crystals -- they are the glue that hold the nuclei together Types of binding (see next slide)

More information

Unit 3 Polynomials Study Guide

Unit 3 Polynomials Study Guide Unit Polynomials Study Guide 7-5 Polynomials Part 1: Classifying Polynomials by Terms Some polynomials have specific names based upon the number of terms they have: # of Terms Name 1 Monomial Binomial

More information

Sample Midterm Solutions

Sample Midterm Solutions Sample Midterm Solutions Instructions: Please answer both questions. You should show your working and calculations for each applicable problem. Correct answers without working will get you relatively few

More information

arxiv:1410.0822v1 [cond-mat.stat-mech] 3 Oct 2014

arxiv:1410.0822v1 [cond-mat.stat-mech] 3 Oct 2014 Reentrance of Berezinskii-Kosterlitz-houless-like transitions in three-state Potts antiferromagnetic thin film Chengxiang Ding, 1, Wenan Guo, 2, and Youjin Deng 3, 1 Department of Applied Physics, Anhui

More information

Field test of a novel combined solar thermal and heat pump system with an ice store

Field test of a novel combined solar thermal and heat pump system with an ice store Field test of a novel combined solar thermal and system with an ice store Anja Loose Institute for Thermodynamics and Thermal Engineering (ITW), Research and Testing Centre for Thermal Solar Systems (TZS),

More information

Fundamentals of Statistical Physics Leo P. Kadanoff University of Chicago, USA

Fundamentals of Statistical Physics Leo P. Kadanoff University of Chicago, USA Fundamentals of Statistical Physics Leo P. Kadanoff University of Chicago, USA text: Statistical Physics, Statics, Dynamics, Renormalization Leo Kadanoff I also referred often to Wikipedia and found it

More information

Dimensional crossover in the non-linear sigma model

Dimensional crossover in the non-linear sigma model INVESTIGACIÓN REVISTA MEXICANA DE FÍSICA 48 (4) 300 306 AGOSTO 00 Dimensional crossover in the non-linear sigma model Denjoe O Connor Departmento de Física, CINVESTAV, IPN Apdo. Post. 4-740, México D.F.

More information

Chapter 29 Scale-Free Network Topologies with Clustering Similar to Online Social Networks

Chapter 29 Scale-Free Network Topologies with Clustering Similar to Online Social Networks Chapter 29 Scale-Free Network Topologies with Clustering Similar to Online Social Networks Imre Varga Abstract In this paper I propose a novel method to model real online social networks where the growing

More information

Quantum Computation with Bose-Einstein Condensation and. Capable of Solving NP-Complete and #P Problems. Abstract

Quantum Computation with Bose-Einstein Condensation and. Capable of Solving NP-Complete and #P Problems. Abstract Quantum Computation with Bose-Einstein Condensation and Capable of Solving NP-Complete and #P Problems Yu Shi Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom Abstract It

More information

The First Law of Thermodynamics

The First Law of Thermodynamics Thermodynamics The First Law of Thermodynamics Thermodynamic Processes (isobaric, isochoric, isothermal, adiabatic) Reversible and Irreversible Processes Heat Engines Refrigerators and Heat Pumps The Carnot

More information

Isotropic Entanglement

Isotropic Entanglement Isotropic Entanglement (Density of States of Quantum Spin Systems) Ramis Movassagh 1 and Alan Edelman 2 1 Department of Mathematics, Northeastern University 2 Department of Mathematics, M.I.T. Fields Institute,

More information

Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization

Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization 2.1. Introduction Suppose that an economic relationship can be described by a real-valued

More information

Definition 11.1. Given a graph G on n vertices, we define the following quantities:

Definition 11.1. Given a graph G on n vertices, we define the following quantities: Lecture 11 The Lovász ϑ Function 11.1 Perfect graphs We begin with some background on perfect graphs. graphs. First, we define some quantities on Definition 11.1. Given a graph G on n vertices, we define

More information

5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1

5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 General Integer Linear Program: (ILP) min c T x Ax b x 0 integer Assumption: A, b integer The integrality condition

More information

Material Deformations. Academic Resource Center

Material Deformations. Academic Resource Center Material Deformations Academic Resource Center Agenda Origin of deformations Deformations & dislocations Dislocation motion Slip systems Stresses involved with deformation Deformation by twinning Origin

More information

Global Seasonal Phase Lag between Solar Heating and Surface Temperature

Global Seasonal Phase Lag between Solar Heating and Surface Temperature Global Seasonal Phase Lag between Solar Heating and Surface Temperature Summer REU Program Professor Tom Witten By Abstract There is a seasonal phase lag between solar heating from the sun and the surface

More information

The Maxwell Demon and Market Efficiency

The Maxwell Demon and Market Efficiency 1 The Maxwell Demon and Market Efficiency Roger D. Jones, Sven G. Redsun, Roger E. Frye, & Kelly D. Myers CommodiCast, Inc. & Complexica, Inc., 125 Lincoln Avenue, Suite 400, Santa Fe, NM 87501 USA This

More information

1 Exercise 3.1b pg 131

1 Exercise 3.1b pg 131 In this solution set, an underline is used to show the last significant digit of numbers. For instance in x = 2.51693 the 2,5,1, and 6 are all significant. Digits to the right of the underlined digit,

More information

Department of Mathematics and Physics (IMFUFA), Roskilde University, Postbox 260, DK-4000, Roskilde, Denmark. E-mail: dyre@ruc.dk

Department of Mathematics and Physics (IMFUFA), Roskilde University, Postbox 260, DK-4000, Roskilde, Denmark. E-mail: dyre@ruc.dk PCCP Computer simulations of the random barrier model Thomas B. Schrøder and Jeppe C. Dyre* Department of Mathematics and Physics (IMFUFA), Roskilde University, Postbox 260, DK-4000, Roskilde, Denmark.

More information

FUNDAMENTALS OF ENGINEERING THERMODYNAMICS

FUNDAMENTALS OF ENGINEERING THERMODYNAMICS FUNDAMENTALS OF ENGINEERING THERMODYNAMICS System: Quantity of matter (constant mass) or region in space (constant volume) chosen for study. Closed system: Can exchange energy but not mass; mass is constant

More information

Minimum requirements for the 13 ka splices for 7 TeV operation

Minimum requirements for the 13 ka splices for 7 TeV operation Minimum requirements for the 13 ka splices for 7 TeV operation - type of defects - FRESCA tests and validation of the code QP3 - a few words on the RRR - I safe vs R addit plots - conclusion Arjan Verweij

More information

Technology of EHIS (stamping) applied to the automotive parts production

Technology of EHIS (stamping) applied to the automotive parts production Laboratory of Applied Mathematics and Mechanics Technology of EHIS (stamping) applied to the automotive parts production Churilova Maria, Saint-Petersburg State Polytechnical University Department of Applied

More information

Cold-Junction-Compensated K-Thermocoupleto-Digital Converter (0 C to +1024 C)

Cold-Junction-Compensated K-Thermocoupleto-Digital Converter (0 C to +1024 C) 19-2235; Rev 1; 3/02 Cold-Junction-Compensated K-Thermocoupleto-Digital General Description The performs cold-junction compensation and digitizes the signal from a type-k thermocouple. The data is output

More information

Backbone and elastic backbone of percolation clusters obtained by the new method of burning

Backbone and elastic backbone of percolation clusters obtained by the new method of burning J. Phys. A: Math. Gen. 17 (1984) L261-L266. Printed in Great Britain LE ITER TO THE EDITOR Backbone and elastic backbone of percolation clusters obtained by the new method of burning H J HerrmanntS, D

More information