V. SCALING. A. Length scale - correlation length. h f 1 L a. (50)
|
|
- Clarissa Holmes
- 5 years ago
- Views:
Transcription
1 V. SCALING A. Length scale - correlation length This broken symmetry business gives rise to a concept pretty much missed until late into the 20th century: length dependence. How big does the system need to be in order for us to truely exhbit the singularities assocciated with a continuous phase transition? The singularity in the magnetization as a function of the magnetic field seems to only be obtained if the system is infinite. When the system is finite we can t observe this singularty, which means that we would see a sudden, but soft, drop in the magnetization as h 0. This drop would sharpen with increasing system size. Now as we approach the transition, the singularity itself - the magnetization jump - becomes smaller. and therefore in order to see this jump, we need to consider bigger and bigger systems. To see this better, let us define h f as the rough symmetry-breaking field, where the magentizationstarts dropping to zero. Clearly, the larger the system, the lower is h f - only at very small SB fields is the system fluctuating and suppressing the long-time average magnetization. So we can guess, that if L is the size of the system, then: h f L a. (50) On the other hand, if we consider h f when the temperature is close to T c, then since even in the infinite system, the average magnetization is tiny, one would need a big h f to stabilize it against finite size fluctuations. So by the same token, we can guess: h f And putting the two observations together, we expect: (T c T ) b (5) h f L a (T c T ) b. (52) Now, suppose we are carrying out the following experiment. We measure the magnetization of several pieces of Ising magnets with different sizes L (but same denisty), but in a constant symmetry breaking field, h 0. For a given temperature T, at what size L, do we start seeing the finite magnetic moments characteristic of the ordered phase? We need to have: h 0 > h f L a (T c T ) b. (53) and therefore: L > (T c T ) = b/a (T c T ) ν (54) We have a special name for this power - ν. This experiment gives an indication of an important length scale that diverges at the critical point. Let s name this mystery length scale, ξ. ξ diverges as we approach the phase transition: ξ t ν (55) where t = (T T c )/T c. form the above discussion, when the size of the system, L > ξ, the singularity of the magnetization is very sharp. But the sharp features of the transition are lost if L < ξ. ξ has an even deeper meaning. It is the correlation length. The rigorous statement associated with the claim above is that the reduced-correlation function decays as: (σ i σ i )(σ j σ j ) e i j /ξ (56) But more important even, is the qualitative meaning. A spin σ i, is sensitive to spins in a sphere of size ξ around it. In order to figure out what it wants to do, it literally seeks the advise of its sphere of interest spins. Pretty much like deciding what stocks to invest in by consulting your best bodies...
2 At the point of the transition, all the spins are so confused, they have to check with everybody in the system, even if it requires asking spins that are miles away. This is the essence of a continuous phase transition. A second qualitative viewpoint on the correlation length, is that it determines the size of droplets that point in the same direction. For instance, above the transition, even though long range order is not established, still there are some flucutations in which a group of spins suddenly form a magnetized droplet. The characteristic size of this droplets is ξ. There is another important conclusion to be made fom this discussion: Whereas in first order phase transitions the system may break down into a coexistence of two phases, second order critical points cannot do that - the size of a droplet diverges at the critical point. A finite size sample will not undergo a sharp transition, but will carry out a croos over between the two phases. 2 B. Critical exponents As it turns out, the critical behavior of systems has many more singularities otehr than that of the correlation length. And eahc singularity is associated with a critical exponent. For teh correlation length, the exponent is ν. Other tabulated singularities:. heat capacity: 2. order parameter: 3. suscptibility: 4. equation of state at t = 0: c t α (57) m t β (58) χ t γ (59) m H /δ (60) the greek letters, here are the critical exponents of the transition. Understanding critical behavior, implies calculating these exponents. VI. UNIVERSALITY, LRO AND MERMIN WAGNER THEOREM, AND EXACT SOLUTION OF THE ISING MODEL A. Role of Models - Universality The greek letters introduced above constitute the critical exponents of a transition. There are other quantities that describe the transition - T c, the speed with which m rises, in fact - the amplitudes in all the above relations. These latter quantities depend very strongly on the microscopic variables of the problem. On the other hand the critical exponents constitute the large-length-scale behavior of the system, and it is independent of the microscopic details. These quantities are therefore called Universal. On the other hand - the quantities that are here are non-universal by definition. As we focus more and more on the large-scale behavior, they fall off, and to us, they are less interesting. For this reason - the independence from microscopic quantities, even the simple bare Ising model is sufficient to descibe the behavior of a critical system that breaks up-down symmetry. This is the essence of Universality. Systems can be categorized into universality classes, which have the same critical behavior (ie critical singularities and exponents), but they differ in the microscopic details of the phases on the two sides of the transition, and in the transition point itself. Systems in the same universality class, share. symmetry,
3 3 2. dimensionality 3. range of interaction (short vs. long with a particular power-law decay) All other details are irrelevant. When I started my graduate studies, many peopled around when discussing physics used the term universal rather freely - they would say - this susceptibility is universal, this is universal, and that is not-universal... and you could tell that they couldn t care less for this last thing. But all this time nobody could tell me what universal actually means. Very quickly i noticed that I don t loose any of the conversation if in my head I translate unviersal, to important. I was quite satisfied with this linguistic short-cut, until it was finally explained to me quite a few years later. Ironically, it was by somebody who wanted to make the point that non-universal aspects fo the system, such as critical temperature, and the amplitude of the diverging quantities is what is actually measured in experiment. I ignored this person. The idea of universality is what makes this quarter worth while - by studying a select group of models, and the canonized methods of scaling and RG, we will be able to understand the behvior of countless systems. B. LRO and QLRO Last class I also emphasized the idea of a broken symmetry and order. How did we qualify order? By defining the order parameter - m = σ (6) when m 0, we have an ordered state. But due to flucutations in finite systems, we know that measuring this equilibrium average might be tricky. An alternative thing we can think of doing, is to ask whether two spin far away from eachother always point in the same direction: if indeed the system is ordered, this correlation will be of the order C i j = S i S j (62) C i j m 2 (63) This we call - Long Range Order. Alternatively order may exist but be fickle, and the correlation will decay very slowly to zero, as a power law: C i j i j φ (64) This is Quasi-long-range order. Long range order is missing if this correlation decays exponentially. In the following, we ll go back to the Ising model and ask whether this LRO we expect at low T can indeed exist. After a qualitative discussion, we will go ahead and just solve the -d Ising exactly. Let s go back to the Ising model: C. Phase transitions at -d? Ĥ = i Jσ i σ j (65) Appealing to your intuition, I claimed that there is a transition temperature above which the Ising model would be disordered and below which it breaks the up down symmetry. But actually is that always true? Suppose it is even in one-d, what do fluctuations do to this symmetry broken state? Consider a very long chain, with all spins pointing up. Disordering such a chain could be most easily done by inserting a domain wall. (DRAW ANOTHER chain). The energy cost of the wall is: 2J
4 The question we need to ask though, is whether the introduction of a domain wall increases or decreases the free energy of the system. ΔF = U T S (66) Since we could insert this domain wall in any of the bonds. This implies an entropy, which is log of the number of possibilities: The free energy of a domain wall, then, is: S = ln L F DW = 2J T ln L (67) Which, as soon as T > 0, becomes. Hence domain walls can always occur - they minimize the free energy. But if domain walls occur freely, then the LRO is completely destroyed. 4 D. 2-d What about 2-d? A domain wall of length L DW now costs roughly U = 2L DW J In addition to being able to occur anywhere along the lattice, which gives a log L contribution, the domain wall can meander as it wishes. roughly speaking it can meander with (z ) L DW possiblities. Therefore the entropy is: S DM ln L + L DW ln(z ) where z is the coordination number of the dual lattice - the number of nearest neighbors in the lattice that the domain wall lives on. This lattice is obtained by drawing lines through the center of the bonds, and perpendicular to them. The log can be neglected compared to the linear term in L DW term. Now the free energy of a domain wall is: which means that below: F L DW (J T ln(z )) (68) T < J/ ln(z ) (69) the Ising model is stable against domain walls. A similar argument applies at higher dimensions. E. Mermin Wagner Thm The above is an illustration of the Mermin-Wagner theorem for the breaking of discrete symmetries - in this case, time reveresal or up-down. Long range order can only obtain at dimensions higher than. We refer to D= as the lower critical dimension of the Ising model. The dimension above which there is no finite temperature symmetry breaking. The Mermin-Wagner theorem implies that this is true for all models with a discrete symmetry - no breaking at finite temperature at D.
5 5 VII. EXACT SOLUTION OF THE ISING MODEL Until now we talked about the general structure of critical phenomena. In particular, we discussed the concept of universality. This concept specifies that critical phenomena is pretty much the same for a variety of models. Therefore, for the rest of the class, we will focus on one particular model - the Ising model, and develop the theory of critical phenomena in this model. We begin with the exact solution of the Ising model, in two setups. First, for free boundaries, and second for periodic boundaries. This way we will gain some intuition and experience as to how interacting models are solved in practice. A. Free boundaries By solution, we mean the partition function and correlations. The partition function is: Z = βj e {S i} L i=0 S is i+ (70) But now, to solve this interacting model, we need a trick. here it is: We can also write as: the last sum is vey easy: = S L =± {S i, i<l} L 2 βj S is i+ e i=0 continue in the same way, and very easily you see that: S L =± e βjs L S L (7) e βjs L S L = e βj + e βj = 2 cosh(βj) (72) Z = (2 cosh(βj)) L (73) real easy, isn t it? Now let me ask you - can you see from here that there is no phase transition? (DISCUSS WITH CLASS) We have to think a bit, but once we have we ralize that in order to have a discontinuity in any order of some thermodynamic derivatice, the partition function need to have a singularity at some point. The partition above, is better behaving the pope himself. The only place where it does something funny is at T = 0, where it diverges. So no criticality, except maybe at T = 0. Above we though of the heat-capacity and susceptibility, for instance. The heat capacity is fairly easy: The entropy: Take another derivative, and you get: F = T ln Z = LT ln(2 cosh(βj)) (74) S = F T = L ln(2 cosh(βj)) L J tanh(βj) (75) T T S ( ) 2 J T = L T cosh 2 βj Near T = 0 we can assume the cosh is just an exponent, and we get: (76) C L ( J T ) 2 e 2βJ (77) Activated decay to zero. The /T 2 looks promising for some divergence, but the exponent just kills it. Also, we know a singularity sohould not be non integrable, otherwise we would have discontinuity in the entropy. so at most: T α with α <. Are we surprised? Not really. The lowest lying excitations are just a domain wall, which cost 2J, and hence the activated form.
6 6 B. Correlation fuctions Unfortunately, we don t know much about correlations from the partiiton function. To calculate the correlations, let me do a trick. The partition function solution is pretty suggestive. It seems as if instead of adding Boltzmann terms associated with each site, we are dealing with eahc bond at a time. Why not then define a bond variable: It is easy to see that: that: η i = S i S i (78) S i = S 0 (S 0 S )(S S 2 )... (S i S i ) = S 0 η η 2... η i (79) and again we use the fact that S 2 i =. Suppose there is one domain wall between 0 and i, this long product will just be. Hence we also call the eta s domain-wall variables. let s call S 0 = η 0 and think about η 0 as a domain wall variable relative to some bounary condition. The partition function is: Z = βj e {S i} L i=0 S is i+ = e βjηi = 2 cosh(βj) (80) so easy! Correlations are also made easy. Consider the correlations between S i and S j wiht j > i. We can write: Now: S i S j = Z {η i} S i S j = e βjηi {η i} j l=i+ j l=i+ η l (8) η l = tanh(βj) i j. (82) This we can write in the form that I showed you earlier - we can write this as an exponential decay with a correlation length: With ξ being: tanh(βj) i j = e i j ln coth(βj) = e i j /ξ (83) ξ = ln coth(βj) = ln( + 2e 2βJ ) 2 e2βj (84) For any finite temperature, we see that ξ is finite, and hence there is no LRO. There is a divergence of ξ as T 0, implying critical behavior there. Usually this divergence is with a power of /t here it is an essential singularity. C. Susceptiblity Even though we didn t manage to solve for the Z with a magnetic field, we can still find the susceptibility. I ll leave it to you to show that: χ = i L h S i = LT L S i S j (85) i,j=0 and to calculate the susceptibility using the above result.
1. Scaling. First carry out a scale transformation, where we zoom out such that all distances appear smaller by a factor b > 1: Δx Δx = Δx/b (307)
45 XVIII. RENORMALIZATION GROUP TECHNIQUE The alternative technique to man field has to be significantly different, and rely on a new idea. This idea is the concept of scaling. Ben Widom, Leo Kadanoff,
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 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 informationPhase transitions and critical phenomena
Phase transitions and critical phenomena Classification of phase transitions. Discontinous (st order) transitions Summary week -5 st derivatives of thermodynamic potentials jump discontinously, e.g. (
More information0.1. CORRELATION LENGTH
0.1. CORRELAION LENGH 0.1 1 Correlation length 0.1.1 Correlation length: intuitive picture Roughly speaking, the correlation length ξ of a spatial configuration is the representative size of the patterns.
More information8.334: Statistical Mechanics II Spring 2014 Test 2 Review Problems
8.334: Statistical Mechanics II Spring 014 Test Review Problems The test is closed book, but if you wish you may bring a one-sided sheet of formulas. The intent of this sheet is as a reminder of important
More informationPhysics 127b: Statistical Mechanics. Second Order Phase Transitions. The Ising Ferromagnet
Physics 127b: Statistical Mechanics Second Order Phase ransitions he Ising Ferromagnet Consider a simple d-dimensional lattice of N classical spins that can point up or down, s i =±1. We suppose there
More informationThe 1+1-dimensional Ising model
Chapter 4 The 1+1-dimensional Ising model The 1+1-dimensional Ising model is one of the most important models in statistical mechanics. It is an interacting system, and behaves accordingly. Yet for a variety
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 informationPhase Transitions and Critical Behavior:
II Phase Transitions and Critical Behavior: A. Phenomenology (ibid., Chapter 10) B. mean field theory (ibid., Chapter 11) C. Failure of MFT D. Phenomenology Again (ibid., Chapter 12) // Windsor Lectures
More informationS i J <ij> h mf = h + Jzm (4) and m, the magnetisation per spin, is just the mean value of any given spin. S i = S k k (5) N.
Statistical Physics Section 10: Mean-Field heory of the Ising Model Unfortunately one cannot solve exactly the Ising model or many other interesting models) on a three dimensional lattice. herefore one
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 informationVI. Series Expansions
VI. Series Expansions VI.A Low-temperature expansions Lattice models can also be studied by series expansions. Such expansions start with certain exactly solvable limits, and typically represent perturbations
More informationPhase transitions and finite-size scaling
Phase transitions and finite-size scaling Critical slowing down and cluster methods. Theory of phase transitions/ RNG Finite-size scaling Detailed treatment: Lectures on Phase Transitions and the Renormalization
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 informationKosterlitz-Thouless Transition
Heidelberg University Seminar-Lecture 5 SS 16 Menelaos Zikidis Kosterlitz-Thouless Transition 1 Motivation Back in the 70 s, the concept of a phase transition in condensed matter physics was associated
More informationOverview of phase transition and critical phenomena
Overview of phase transition and critical phenomena Aims: Phase transitions are defined, and the concepts of order parameter and spontaneously broken symmetry are discussed. Simple models for magnetic
More informationPhase transition and spontaneous symmetry breaking
Phys60.nb 111 8 Phase transition and spontaneous symmetry breaking 8.1. Questions: Q1: Symmetry: if a the Hamiltonian of a system has certain symmetry, can the system have a lower symmetry? Q: Analyticity:
More informationStatistical Thermodynamics Solution Exercise 8 HS Solution Exercise 8
Statistical Thermodynamics Solution Exercise 8 HS 05 Solution Exercise 8 Problem : Paramagnetism - Brillouin function a According to the equation for the energy of a magnetic dipole in an external magnetic
More informationPhysics Nov Phase Transitions
Physics 301 11-Nov-1999 15-1 Phase Transitions Phase transitions occur throughout physics. We are all familiar with melting ice and boiling water. But other kinds of phase transitions occur as well. Some
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 informationClusters and Percolation
Chapter 6 Clusters and Percolation c 2012 by W. Klein, Harvey Gould, and Jan Tobochnik 5 November 2012 6.1 Introduction In this chapter we continue our investigation of nucleation near the spinodal. We
More informationLab 70 in TFFM08. Curie & Ising
IFM The Department of Physics, Chemistry and Biology Lab 70 in TFFM08 Curie & Ising NAME PERS. -NUMBER DATE APPROVED Rev Aug 09 Agne 1 Introduction Magnetic materials are all around us, and understanding
More informationPhase Transitions and Renormalization:
Phase Transitions and Renormalization: Using quantum techniques to understand critical phenomena. Sean Pohorence Department of Applied Mathematics and Theoretical Physics University of Cambridge CAPS 2013
More informationVI.D Self Duality in the Two Dimensional Ising Model
VI.D Self Duality in the Two Dimensional Ising Model Kramers and Wannier discovered a hidden symmetry that relates the properties of the Ising model on the square lattice at low and high temperatures.
More informationThe (magnetic) Helmholtz free energy has proper variables T and B. In differential form. and the entropy and magnetisation are thus given by
4.5 Landau treatment of phase transitions 4.5.1 Landau free energy In order to develop a general theory of phase transitions it is necessary to extend the concept of the free energy. For definiteness we
More informationVI.D Self Duality in the Two Dimensional Ising Model
VI.D Self Duality in the Two Dimensional Ising Model Kramers and Wannier discovered a hidden symmetry that relates the properties of the Ising model on the square lattice at low and high temperatures.
More informationNumerical Analysis of 2-D Ising Model. Ishita Agarwal Masters in Physics (University of Bonn) 17 th March 2011
Numerical Analysis of 2-D Ising Model By Ishita Agarwal Masters in Physics (University of Bonn) 17 th March 2011 Contents Abstract Acknowledgment Introduction Computational techniques Numerical Analysis
More informationφ(ν)dν = 1. (1) We can define an average intensity over this profile, J =
Ask about final Saturday, December 14 (avoids day of ASTR 100 final, Andy Harris final). Decided: final is 1 PM, Dec 14. Rate Equations and Detailed Balance Blackbodies arise if the optical depth is big
More information8 Error analysis: jackknife & bootstrap
8 Error analysis: jackknife & bootstrap As discussed before, it is no problem to calculate the expectation values and statistical error estimates of normal observables from Monte Carlo. However, often
More informationEnergy is always partitioned into the maximum number of states possible.
ENTROPY Entropy is another important aspect of thermodynamics. Enthalpy has something to do with the energetic content of a system or a molecule. Entropy has something to do with how that energy is stored.
More informationPhase transitions beyond the Landau-Ginzburg theory
Phase transitions beyond the Landau-Ginzburg theory Yifei Shi 21 October 2014 1 Phase transitions and critical points 2 Laudau-Ginzburg theory 3 KT transition and vortices 4 Phase transitions beyond Laudau-Ginzburg
More informationPhase Transitions: A Challenge for Reductionism?
Phase Transitions: A Challenge for Reductionism? Patricia Palacios Munich Center for Mathematical Philosophy Abstract In this paper, I analyze the extent to which classical phase transitions, especially
More informationThe Ising model Summary of L12
The Ising model Summary of L2 Aim: Study connections between macroscopic phenomena and the underlying microscopic world for a ferromagnet. How: Study the simplest possible model of a ferromagnet containing
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 informationBrief review of Quantum Mechanics (QM)
Brief review of Quantum Mechanics (QM) Note: This is a collection of several formulae and facts that we will use throughout the course. It is by no means a complete discussion of QM, nor will I attempt
More informationThe Phase Transition of the 2D-Ising Model
The Phase Transition of the 2D-Ising Model Lilian Witthauer and Manuel Dieterle Summer Term 2007 Contents 1 2D-Ising Model 2 1.1 Calculation of the Physical Quantities............... 2 2 Location of the
More informationPart I Electrostatics. 1: Charge and Coulomb s Law July 6, 2008
Part I Electrostatics 1: Charge and Coulomb s Law July 6, 2008 1.1 What is Electric Charge? 1.1.1 History Before 1600CE, very little was known about electric properties of materials, or anything to do
More informationPotts And XY, Together At Last
Potts And XY, Together At Last Daniel Kolodrubetz Massachusetts Institute of Technology, Center for Theoretical Physics (Dated: May 16, 212) We investigate the behavior of an XY model coupled multiplicatively
More informationf(t,h) = t 2 g f (h/t ), (3.2)
Chapter 3 The Scaling Hypothesis Previously, we found that singular behaviour in the vicinity of a second order critical point was characterised by a set of critical exponents {α,β,γ,δ, }. These power
More informationSpin glasses and Adiabatic Quantum Computing
Spin glasses and Adiabatic Quantum Computing A.P. Young alk at the Workshop on heory and Practice of Adiabatic Quantum Computers and Quantum Simulation, ICP, rieste, August 22-26, 2016 Spin Glasses he
More informationPhysics 127b: Statistical Mechanics. Renormalization Group: 1d Ising Model. Perturbation expansion
Physics 17b: Statistical Mechanics Renormalization Group: 1d Ising Model The ReNormalization Group (RNG) gives an understanding of scaling and universality, and provides various approximation schemes to
More informationIntroductory Quantum Chemistry Prof. K. L. Sebastian Department of Inorganic and Physical Chemistry Indian Institute of Science, Bangalore
Introductory Quantum Chemistry Prof. K. L. Sebastian Department of Inorganic and Physical Chemistry Indian Institute of Science, Bangalore Lecture - 4 Postulates Part 1 (Refer Slide Time: 00:59) So, I
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 informationScaling Theory. Roger Herrigel Advisor: Helmut Katzgraber
Scaling Theory Roger Herrigel Advisor: Helmut Katzgraber 7.4.2007 Outline The scaling hypothesis Critical exponents The scaling hypothesis Derivation of the scaling relations Heuristic explanation Kadanoff
More informationInteracting Fermi Gases
Interacting Fermi Gases Mike Hermele (Dated: February 11, 010) Notes on Interacting Fermi Gas for Physics 7450, Spring 010 I. FERMI GAS WITH DELTA-FUNCTION INTERACTION Since it is easier to illustrate
More informationMIT BLOSSOMS INITIATIVE
MIT BLOSSOMS INITIATIVE The Broken Stick Problem Taught by Professor Richard C. Larson Mitsui Professor of Engineering Systems and of Civil and Environmental Engineering Segment 1 Hi! My name is Dick Larson
More informationSums of Squares (FNS 195-S) Fall 2014
Sums of Squares (FNS 195-S) Fall 014 Record of What We Did Drew Armstrong Vectors When we tried to apply Cartesian coordinates in 3 dimensions we ran into some difficulty tryiing to describe lines and
More informationFigure 1: Doing work on a block by pushing it across the floor.
Work Let s imagine I have a block which I m pushing across the floor, shown in Figure 1. If I m moving the block at constant velocity, then I know that I have to apply a force to compensate the effects
More informationLecture 10: Powers of Matrices, Difference Equations
Lecture 10: Powers of Matrices, Difference Equations Difference Equations A difference equation, also sometimes called a recurrence equation is an equation that defines a sequence recursively, i.e. each
More informationIn economics, the amount of a good x demanded is a function of the price of that good. In other words,
I. UNIVARIATE CALCULUS Given two sets X and Y, a function is a rule that associates each member of X with eactly one member of Y. That is, some goes in, and some y comes out. These notations are used to
More informationMetropolis Monte Carlo simulation of the Ising Model
Metropolis Monte Carlo simulation of the Ising Model Krishna Shrinivas (CH10B026) Swaroop Ramaswamy (CH10B068) May 10, 2013 Modelling and Simulation of Particulate Processes (CH5012) Introduction The Ising
More informationMechanics and Statistical Mechanics Qualifying Exam Spring 2006
Mechanics and Statistical Mechanics Qualifying Exam Spring 2006 1 Problem 1: (10 Points) Identical objects of equal mass, m, are hung on identical springs of constant k. When these objects are displaced
More informationPhase Transitions. µ a (P c (T ), T ) µ b (P c (T ), T ), (3) µ a (P, T c (P )) µ b (P, T c (P )). (4)
Phase Transitions A homogeneous equilibrium state of matter is the most natural one, given the fact that the interparticle interactions are translationally invariant. Nevertheless there is no contradiction
More informationA Simple Model s Best Hope: A Brief Introduction to Universality
A Simple Model s Best Hope: A Brief Introduction to Universality Benjamin Good Swarthmore College (Dated: May 5, 2008) For certain classes of systems operating at a critical point, the concept of universality
More informationX. Assembling the Pieces
X. Assembling the Pieces 179 Introduction Our goal all along has been to gain an understanding of nuclear reactors. As we ve noted many times, this requires knowledge of how neutrons are produced and lost.
More informationSolving with Absolute Value
Solving with Absolute Value Who knew two little lines could cause so much trouble? Ask someone to solve the equation 3x 2 = 7 and they ll say No problem! Add just two little lines, and ask them to solve
More informationLuigi Paolasini
Luigi Paolasini paolasini@esrf.fr LECTURE 7: Magnetic excitations - Phase transitions and the Landau mean-field theory. - Heisenberg and Ising models. - Magnetic excitations. External parameter, as for
More informationE = <ij> N 2. (0.3) (so that k BT
Phys 4 Project solutions. November 4, 7 Note the model and units Quantities The D Ising model has the total energy E written in terms of the spins σ i = ± site i, E = σ i σ j (.) where < ij > indicates
More informationRenormalization Group for the Two-Dimensional Ising Model
Chapter 8 Renormalization Group for the Two-Dimensional Ising Model The two-dimensional (2D) Ising model is arguably the most important in statistical physics. This special status is due to Lars Onsager
More informationNon-magnetic states. The Néel states are product states; φ N a. , E ij = 3J ij /4 2 The Néel states have higher energy (expectations; not eigenstates)
Non-magnetic states Two spins, i and j, in isolation, H ij = J ijsi S j = J ij [Si z Sj z + 1 2 (S+ i S j + S i S+ j )] For Jij>0 the ground state is the singlet; φ s ij = i j i j, E ij = 3J ij /4 2 The
More informationΩ = e E d {0, 1} θ(p) = P( C = ) So θ(p) is the probability that the origin belongs to an infinite cluster. It is trivial that.
2 Percolation There is a vast literature on percolation. For the reader who wants more than we give here, there is an entire book: Percolation, by Geoffrey Grimmett. A good account of the recent spectacular
More informationSniffing out new laws... Question
Sniffing out new laws... How can dimensional analysis help us figure out what new laws might be? (Why is math important not just for calculating, but even just for understanding?) (And a roundabout way
More informationWhat is proof? Lesson 1
What is proof? Lesson The topic for this Math Explorer Club is mathematical proof. In this post we will go over what was covered in the first session. The word proof is a normal English word that you might
More informationThis is the third of three lectures on cavity theory.
This is the third of three lectures on cavity theory. 1 In this lecture, we are going to go over what is meant by charged particle equilibrium and look at the dose and kerma when you have charged particle
More informationIsing Model. Ising Lattice. E. J. Maginn, J. K. Shah
Ising Lattice E. J. Maginn, J. K. Shah Department of Chemical and Biomolecular Engineering University of Notre Dame Notre Dame, IN 46556 USA Monte Carlo Workshop, Brazil Ising Lattice Model Consider a
More informationDEVELOPING MATH INTUITION
C HAPTER 1 DEVELOPING MATH INTUITION Our initial exposure to an idea shapes our intuition. And our intuition impacts how much we enjoy a subject. What do I mean? Suppose we want to define a cat : Caveman
More informationLecture 4: Constructing the Integers, Rationals and Reals
Math/CS 20: Intro. to Math Professor: Padraic Bartlett Lecture 4: Constructing the Integers, Rationals and Reals Week 5 UCSB 204 The Integers Normally, using the natural numbers, you can easily define
More informationMATH 230 CALCULUS II OVERVIEW
MATH 230 CALCULUS II OVERVIEW This overview is designed to give you a brief look into some of the major topics covered in Calculus II. This short introduction is just a glimpse, and by no means the whole
More informationI Basic Spin Physics. 1. Nuclear Magnetism
I Basic Spin Physics Lecture notes by Assaf Tal The simplest example of a magnetic moment is the refrigerator magnet. We ll soon meet other, much smaller and weaker magnetic moments, when we discuss the
More informationLecture 11: Long-wavelength expansion in the Neel state Energetic terms
Lecture 11: Long-wavelength expansion in the Neel state Energetic terms In the last class we derived the low energy effective Hamiltonian for a Mott insulator. This derivation is an example of the kind
More informationIntroduction to the Renormalization Group
Introduction to the Renormalization Group Gregory Petropoulos University of Colorado Boulder March 4, 2015 1 / 17 Summary Flavor of Statistical Physics Universality / Critical Exponents Ising Model Renormalization
More informationEquilibrium, out of equilibrium and consequences
Equilibrium, out of equilibrium and consequences Katarzyna Sznajd-Weron Institute of Physics Wroc law University of Technology, Poland SF-MTPT Katarzyna Sznajd-Weron (WUT) Equilibrium, out of equilibrium
More informationMonte Carlo Simulation of the Ising Model. Abstract
Monte Carlo Simulation of the Ising Model Saryu Jindal 1 1 Department of Chemical Engineering and Material Sciences, University of California, Davis, CA 95616 (Dated: June 9, 2007) Abstract This paper
More informationPHYS Statistical Mechanics I Course Outline
PHYS 449 - Statistical Mechanics I Course Outline There is no official textbook for this course. Suggested References: An Introduction to Thermal Physics, by Daniel V. Schroeder: this was the textbook
More informationPhase Transitions in Spin Glasses
p.1 Phase Transitions in Spin Glasses Peter Young http://physics.ucsc.edu/ peter/talks/bifi2008.pdf e-mail:peter@physics.ucsc.edu Work supported by the and the Hierarchical Systems Research Foundation.
More informationSlightly off-equilibrium dynamics
Slightly off-equilibrium dynamics Giorgio Parisi Many progresses have recently done in understanding system who are slightly off-equilibrium because their approach to equilibrium is quite slow. In this
More informationPhase Transitions: A Challenge for Reductionism?
Phase Transitions: A Challenge for Reductionism? Patricia Palacios Munich Center for Mathematical Philosophy Abstract In this paper, I analyze the extent to which classical phase transitions, especially
More informationAtomic Theory. Introducing the Atomic Theory:
Atomic Theory Chemistry is the science of matter. Matter is made up of things called atoms, elements, and molecules. But have you ever wondered if atoms and molecules are real? Would you be surprised to
More informationGuide to Proofs on Sets
CS103 Winter 2019 Guide to Proofs on Sets Cynthia Lee Keith Schwarz I would argue that if you have a single guiding principle for how to mathematically reason about sets, it would be this one: All sets
More informationHi, my name is Dr. Ann Weaver of Argosy University. This WebEx is about something in statistics called z-
Hi, my name is Dr. Ann Weaver of Argosy University. This WebEx is about something in statistics called z- Scores. I have two purposes for this WebEx, one, I just want to show you how to use z-scores in
More informationInstructor (Brad Osgood)
TheFourierTransformAndItsApplications-Lecture26 Instructor (Brad Osgood): Relax, but no, no, no, the TV is on. It's time to hit the road. Time to rock and roll. We're going to now turn to our last topic
More informationThe glass transition as a spin glass problem
The glass transition as a spin glass problem Mike Moore School of Physics and Astronomy, University of Manchester UBC 2007 Co-Authors: Joonhyun Yeo, Konkuk University Marco Tarzia, Saclay Mike Moore (Manchester)
More informationPhys Midterm. March 17
Phys 7230 Midterm March 17 Consider a spin 1/2 particle fixed in space in the presence of magnetic field H he energy E of such a system can take one of the two values given by E s = µhs, where µ is the
More informationCritical Behavior I: Phenomenology, Universality & Scaling
Critical Behavior I: Phenomenology, Universality & Scaling H. W. Diehl Fachbereich Physik, Universität Duisburg-Essen, Campus Essen 1 Goals recall basic facts about (static equilibrium) critical behavior
More informationPhysics I: Oscillations and Waves Prof. S. Bharadwaj Department of Physics and Meteorology. Indian Institute of Technology, Kharagpur
Physics I: Oscillations and Waves Prof. S. Bharadwaj Department of Physics and Meteorology Indian Institute of Technology, Kharagpur Lecture No 03 Damped Oscillator II We were discussing, the damped oscillator
More informationSection B. Electromagnetism
Prelims EM Spring 2014 1 Section B. Electromagnetism Problem 0, Page 1. An infinite cylinder of radius R oriented parallel to the z-axis has uniform magnetization parallel to the x-axis, M = m 0ˆx. Calculate
More informationModern Physics notes Spring 2007 Paul Fendley Lecture 27
Modern Physics notes Spring 2007 Paul Fendley fendley@virginia.edu Lecture 27 Angular momentum and positronium decay The EPR paradox Feynman, 8.3,.4 Blanton, http://math.ucr.edu/home/baez/physics/quantum/bells
More informationPhysics 127b: Statistical Mechanics. Landau Theory of Second Order Phase Transitions. Order Parameter
Physics 127b: Statistical Mechanics Landau Theory of Second Order Phase Transitions Order Parameter Second order phase transitions occur when a new state of reduced symmetry develops continuously from
More informationSlope Fields: Graphing Solutions Without the Solutions
8 Slope Fields: Graphing Solutions Without the Solutions Up to now, our efforts have been directed mainly towards finding formulas or equations describing solutions to given differential equations. Then,
More informationIntroduction to Algebra: The First Week
Introduction to Algebra: The First Week Background: According to the thermostat on the wall, the temperature in the classroom right now is 72 degrees Fahrenheit. I want to write to my friend in Europe,
More informationModern Physics notes Paul Fendley Lecture 1
Modern Physics notes Paul Fendley fendley@virginia.edu Lecture 1 What is Modern Physics? Topics in this Class Books Their Authors Feynman 1.1 What is Modern Physics? This class is usually called modern
More informationQ: How can quantum computers break ecryption?
Q: How can quantum computers break ecryption? Posted on February 21, 2011 by The Physicist Physicist: What follows is the famous Shor algorithm, which can break any RSA encryption key. The problem: RSA,
More informationA Monte Carlo Implementation of the Ising Model in Python
A Monte Carlo Implementation of the Ising Model in Python Alexey Khorev alexey.s.khorev@gmail.com 2017.08.29 Contents 1 Theory 1 1.1 Introduction...................................... 1 1.2 Model.........................................
More informationRenormalization Group: non perturbative aspects and applications in statistical and solid state physics.
Renormalization Group: non perturbative aspects and applications in statistical and solid state physics. Bertrand Delamotte Saclay, march 3, 2009 Introduction Field theory: - infinitely many degrees of
More information3.091 Introduction to Solid State Chemistry. Lecture Notes No. 6a BONDING AND SURFACES
3.091 Introduction to Solid State Chemistry Lecture Notes No. 6a BONDING AND SURFACES 1. INTRODUCTION Surfaces have increasing importance in technology today. Surfaces become more important as the size
More informationUnstable K=K T=T. Unstable K=K T=T
G252651: Statistical Mechanics Notes for Lecture 28 I FIXED POINTS OF THE RG EQUATIONS IN GREATER THAN ONE DIMENSION In the last lecture, we noted that interactions between block of spins in a spin lattice
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 informationQuantum Annealing in spin glasses and quantum computing Anders W Sandvik, Boston University
PY502, Computational Physics, December 12, 2017 Quantum Annealing in spin glasses and quantum computing Anders W Sandvik, Boston University Advancing Research in Basic Science and Mathematics Example:
More information(1) Consider a lattice of noninteracting spin dimers, where the dimer Hamiltonian is
PHYSICS 21 : SISICL PHYSICS FINL EXMINION 1 Consider a lattice of noninteracting spin dimers where the dimer Hamiltonian is Ĥ = H H τ τ Kτ where H and H τ are magnetic fields acting on the and τ spins
More information