DOCSLIB.ORG

Phase Transitions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Phase Transitions Phase Transitions Proseminar in Theoretical Physics Institut fur¨ theoretische Physik ETH Zurich¨ SS07 typeset & compilation by H. G. Katzgraber (with help from F. Hassler) Table of Contents 1 Symmetry and ergodicity breaking, mean-field study of the Ising model 1 1.1 Introduction.............................. 1 1.2 Formalism............................... 2 1.3 Themodelsystem:Isingmodel . 4 1.4 Solutionsinoneandmoredimensions. 8 1.5 Symmetryandergodicitybreaking . 14 1.6 Concludingremarks. 17 2 Critical Phenomena in Fluids 21 2.1 Repetition of the needed Thermodynamics . 21 2.2 Two-phaseCoexistence. 23 2.3 Van der Waals equation . 26 2.4 Spatialcorrelations . 30 2.5 Determination of the critical point/ difficulties in measurement . 32 3 Landau Theory, Fluctuations and Breakdown of Landau The- ory 37 3.1 Introduction.............................. 37 3.2 LandauTheory ............................ 40 3.3 Fluctuations.............................. 44 3.4 ConclusiveWords........................... 49 4 Scaling theory 53 4.1 Criticalexponents........................... 53 4.2 Scalinghypothesis .......................... 54 4.3 Deriving the Homogeneous forms . 58 4.4 Universalityclasses .......................... 61 4.5 Polymerstatistics........................... 62 4.6 Conclusions .............................. 68 iii TABLE OF CONTENTS 5 The Renormalization Group 71 5.1 Introduction.............................. 71 5.2 Classical lattice spin systems . 72 5.3 Scalinglimit.............................. 73 5.4 Renormalization group transformations . 76 5.5 Example: The Gaussian fixed point . 83 6 Critical slowing down and cluster updates in Monte Carlo simulations 89 6.1 Introduction.............................. 89 6.2 Criticalslowingdown. 92 6.3 Clusterupdates ............................ 95 6.4 Concludingremarks. 101 7 Finite-Size Scaling 105 7.1 Introduction.............................. 105 7.2 Scaling Function Hypothesis . 106 7.3 Critical Temperature Tc ....................... 109 7.4 Distinguishing between 1st and 2nd Order Phase Transitions . 113 7.5 Concludingremarks. 114 8 Berezinskii-Kosterlitz-Thouless Transition 117 8.1 Introduction.............................. 117 8.2 Vortices ................................ 119 8.3 Villain and Coulomb gas models . 121 8.4 Effectiveinteraction. 128 8.5 Kosterlitz-Thouless Transition . 131 9 O(N) Model and 1/N Expansion 137 9.1 Introduction.............................. 137 9.2 TheModel............................... 138 9.3 Mathematical Background . 139 9.4 SpontaneousSymmetryBreaking . 142 9.5 1/Nexpansion............................. 149 9.6 Renormalization and critical exponents . 152 9.7 Concludingremarks. 154 10 Quantum Phase Transitions and the Bose-Hubbard Model 157 10.1 What is a Quantum Phase Transition? . 157 10.2 TheBose-HubbardModel . 160 iv TABLE OF CONTENTS 10.3 Ultra-cold bosonic Gas in an optical Lattice Potential . 169 11 Quantum Field Theory and the Deconfinement Transition 175 11.1 Introduction.............................. 175 11.2 GaugeFieldTheories . 176 11.3 Wilson Loop and Static qq-Potential . 177 11.4 Quantum Field Theories at Finite Temperature . 180 11.5 The Wilson Line or Polyakov Loop . 183 11.6 The Deconfinement Phase Transition . 184 11.7 Deconfinement in the SU(3)GaugeTheory. 188 12 The Phases of Quantum Chromodynamics 193 12.1 Introduction.............................. 193 12.2 TheQCDPhaseDiagram . 203 12.3 Colour superconductivity and colour flavourlocking............................. 211 12.4 Relativistic Heavy-Ion Collisions . 215 13 BCS theory of superconductivity 221 13.1 Introduction.............................. 221 13.2CooperPairs ............................. 222 13.3BCSTheory.............................. 224 13.4 FiniteTemperatures . 228 13.5 Type II Superconductors and the Vortex State . 233 14 Bose Einstein Condensation 239 14.1 Introduction.............................. 239 14.2 The non-interacting Bose gas . 240 14.3 The Bose Gas with Hard Cores in the Mean Field Limit . 243 14.4 The weakly interacting Bose Gas in the Critical Regime . 245 14.5 Spontaneous Symmetry Breaking in the Bose Gas . 247 15 Superfluidity and Supersolidity 251 15.1 SuperfluidHelium. 251 15.2 SupersolidHelium. 260 v TABLE OF CONTENTS vi Chapter 1 Symmetry and ergodicity breaking, mean-field study of the Ising model Rafael Mottl supervisor: Dr. Ingo Kirsch We describe a ferromagnetic domain in a magnetic material by the Ising model using the language of statistical mechanics. By heuristic arguments we show that there is no spontaneous magne- tization for the nearest-neighbour Ising model in one dimension and that there is spontaneous magnetization in two and three dimensions. Three solutions are presented: the transfer matrix method in one dimension and non-zero external magnetic field, some ideas of the Onsager solution for the two-dimensional case with zero external magnetic field and the mean-field theory for arbitrary dimension and non-zero external magnetic field. Sym- metry and ergodicity breaking are introduced and applied to the Ising model. The restricted ensemble is used to describe the sys- tem with broken ergodicity. 1.1 Introduction We describe a ferromagnetic domain with spontaneous magnetization. In a cor- related state of spontaneous magnetization we have an ordered system whereas above some critical temperature we observe an unordered state: a phase transi- tion occurs at the critical temperature TC . 1 1.2 Formalism This phase transition is described by an order parameter which is zero above the critical temperature (zero order) and non-zero below the critical temperature: a measure of order which is in our case of a magnetic material the magnetization M of the material. From a general Hamiltonian with all possible spin-interaction terms we make a couple of assumptions to get to the Ising model. This model was introduced by the German physicist Ernst Ising (1900 - 1998). 1.2 Formalism The analysis of the Ising model makes use of the statistical mechanical descrip- tion of the system. We remember the reader of the statistical mechanical basics to introduce afterwards the definitions of phases, phase transition and critical exponents. 1.2.1 Statistical mechanical basics We describe a certain sample region Ω of volume V (Ω) which contains a number of particles N(Ω). The analysis is based on the canonical ensemble description. The corresponding partition function is given by: βH ( Kn , Θn ) Z [ K ] = Tr exp − Ω { } { } , (1.1) Ω { n} 1 β = , (1.2) kBT where the trace operator is the sum over all possible states in phase space Γ. The Hamiltonian defined on the phase space depends on the degrees of freedom Θn of the system (in our case: spin variables) and certain coupling constants Kn (e.g. external magnetic field). From the partition function we can calculate the free energy F [ K ]= F [K]= k T log Z [K]. (1.3) Ω { n} Ω − B Ω By taking the thermodynamical limit we get the bulk free energy per site FΩ[K] fb[K] = lim . (1.4) N(Ω) N(Ω) →∞ As we will see below, phase boundaries are defined as the regions of non-analyticity of the bulk free energy per site. We need the thermodynamic limit in order to 2 Symmetry and ergodicity breaking, mean-field study of the Ising model describe phase transitions. As the bulk free energy per site is just a finite sum over exponential terms having non-singular exponents it would be analytic every- where. Thus phase transitions are only observed in the thermodynamical limit. By applying partial derivatives to the bulk free energy we obtain the well-known thermodynamical properties: internal energy, heat capacity, magnetization and magnetic susceptibility: ∂ ǫ [K]= (βf [K]), (1.5) in ∂β b ∂ǫ [K] C[K]= in , (1.6) ∂T ∂f [K] M[K]= b , (1.7) − ∂H ∂M[K] χ [K]= . (1.8) T ∂H 1.2.2 Phases, phase transitions and critical exponents A phase is a region of analyticity of the bulk free energy per site. Different phases are separated by phase boundaries which are singular loci of the bulk free energy per site of dimension d - 1. We distinguish between two different kinds of phase transitions: 1. first-order phase transition: one (or more) of the first partial derivatives of the bulk free energy per site is discontinuous 2. continuous phase transition: all first derivatives of the bulk free energy per site are continuous. For a continuous phase transition we can introduce the critical exponents which describe the behaviour of the thermodynamical properties near the phase transi- tion: α 1. heat capacity C t − ∼| | 2. order parameter M t β ∼| | γ 3. susceptibility χ t − ∼| | 3 1.3 The model system: Ising model 4. equation of state M H1/δ. ∼ The first three critical exponents are defined for zero external magnetic field and t describes the deviation of the temperature from the critical temperature: T T t = − C . (1.9) TC The notation means that the described value has a singular part proportional ∼ to the given value. 1.2.3 Correlation function, correlation length and their critical exponents The definition of the correlation function looks as follows: G(r) := (S(r) S(r) ) (S(0) S(0) ) , (1.10) −h i −h i where r is the location of the spin variable S(r). The correlation function mea- sures the correlation between the fluctuations of the spin variables S(r) and S(0). Near the critical temperature TC we assume that the correlation function has the so called Ornstein-Zernike form p r/ξ G(r) r− e− . (1.11) ≃ ξ is the correlation length, i.e. the “length scale over which the fluctuations of the microscopic degrees of freedom are significantly correlated with each other” [1]. Cooling the system down near the critical temperature, we observe the divergence of the correlation length. Finally, the critical exponents ν and η for the correlation function are defined as ν ξ t − , p = d 2+ η. (1.12) ∼| | − 1.3 The model system: Ising model We introduce the Ising model and want to discuss the possibility of observing spontaneous magnetization in different dimensions. 4 Symmetry and ergodicity breaking, mean-field study of the Ising model 1.3.1 Characterization of the Ising model The properties of the general spin system are the follwing: 1. periodic lattice Ω in d dimensions 2. lattice contains N(Ω) fixed points called lattice sites 3.
Load more

Recommended publications
  • Advanced Statistical Physics
    UniversityofCologne InstituteforTheoreticalPhysics Advanced Statistical Physics Lectures: Johannes Berg Exercises: Simon Dettmer and Chau Nguyen Sheet 10: Christmas review Due: No hand-in required (Mailbox: Advanced Statistical Physics) Discussion: 12:00-13:30 Monday, 13 January, 2014 20. An intermediate review This is a good time to look back at what we have archived. The short questions below are intended to help you review some of the important concepts, and link them to some historical developments. For a reference, we recommend that you read N. Goldenfeld, “Lectures on phase transitions and the renormalisation group”, chapter 1 (Addison-Wesley Publishing Company, 1992), if you have not yet done so. At the beginning of the course, we started with a discussion on the derivation of the Boltz- mann distribution based understandings in stochastic processes. We sketched the conditions under which the Boltzmann distributions emerge as an invariant measure of a stochastic pro- cess. A by-product of the discussion was Markov chain Monte Carlo (MCMC) simulation, which is a very important technique well beyond statistical mechanics. a) Can you give a physical example of a stochastic system that does not obey the Boltzmann distribution? We further illustrated the method of partition functions in statistical mechanics, Ising models were taken as paradigms (Ising models are said to be the physicists’ Drosophila). We showed that the partition functions of one-dimensional systems are usually solvable by the recursion method or the transfer matrix method. The method of low temperature series and high temperature series were also introduced, which found very important applications later on. When we studied the two-dimensional Ising model, for the first time (in this course) we encountered the concept of phase transition.
    [Show full text]
  • Uwaterloo Latex Thesis Template
    Explorations in black hole chemistry and higher curvature gravity by Robie Hennigar A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Physics Waterloo, Ontario, Canada, 2018 c Robie Hennigar 2018 Examining Committee Membership The following served on the Examining Committee for this thesis. The decision of the Examining Committee is by majority vote. External Examiner: Don Page Professor, University of Alberta Supervisor: Robert Mann Professor, University of Waterloo Internal Member: Niayesh Afshordi Professor, University of Waterloo Internal-External Member: Achim Kempf Professor, University of Waterloo Committee Member: Robert Myers Professor, Perimeter Institute for Theoretical Physics, University of Waterloo ii This thesis consists of material all of which I authored or co-authored: see Statement of Contributions included in the thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Statement of contributions This thesis is based on the following published and forthcoming articles. • Chapter3 is based on: { R. A. Hennigar, D. Kubiznak and R. B. Mann, Entropy Inequality Violations from Ultraspinning Black Holes, Phys. Rev. Lett. 115 (2015) 031101, [1411.4309] { R. A. Hennigar, D. Kubiznak, R. B. Mann and N. Musoke, Ultraspinning limits and super-entropic black holes, JHEP 06 (2015) 096, [1504.07529] { R. A. Hennigar, D. Kubiznak, R. B. Mann and N. Musoke, Ultraspinning lim- its and rotating hyperboloid membranes, Nucl. Phys. B903 (2016) 400{417, [1512.02293] • Chapter4 is based on: { R.
    [Show full text]
  • Table of Contents More Information
    Cambridge University Press 978-1-107-15401-8 — Perspectives on Statistical Thermodynamics Yoshitsugu Oono Table of Contents More Information Contents Preface page xxi Acknowledgements xxiii Self-Study Guide xxiv List of Symbols xxvi Part I Atomic and Mesoscopic Viewpoint 1 1Introduction 3 1.1 Atomisms, Ancient and Modern 4 1.2 What was beyond Philosophers’ Grasp? 4 1.3 How Numerous are Atoms and Molecules? 5 1.4 Why are Molecules so Small? 5 1.5 Our World is Lawful to the Extent of Allowing the Evolution of Intelligence 6 1.6 Microscopic World is Unpredictable 7 1.7 Why our Macroscopic World is Lawful: the Law of Large Numbers 7 1.8 We Live in a Rather Gentle World 8 1.9 Thermodynamics, Statistical Mechanics, and Phase Transition 9 1.10 The Mesoscopic World 10 1.11 Large Deviation and Fluctuation 10 Problems 11 2 Atomic Picture of Gases 12 2.1 Aristotelian Physics and Galileo’s Struggle 12 2.2 Boyle: The True Pioneer of Kinetic Theory of Heat 13 2.3 Discovery of Atmospheric Pressure and Vacuum 13 2.4 Daniel Bernoulli’s Modern Dynamic Atomism and a 100 Year Hiatus 14 2.5 Between Bernoulli and Maxwell 14 2.6 Daniel Bernoulli’s Kinetic Theory 15 2.7 Equipartition of Kinetic Energy 17 Problems 18 3 Introduction to Probability 20 3.1 Probability is a Measure of Confidence Level 21 3.2 Events and Sets 21 vii © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-15401-8 — Perspectives on Statistical Thermodynamics Yoshitsugu Oono Table of Contents More Information viii Contents 3.3 Probability
    [Show full text]
  • Critical Exponents for Schwarzschild-Kerr and BTZ Systems
    Critical Exponents For Schwarzschild-Kerr and BTZ Systems J.P. Krisch and E.N. Glass∗ Department of Physics, University of Michigan, Ann Arbor, Michgan 48109 (12 February 2001, preprint MCTP-01-08) Regarding the spin-up of Schwarzschild-Kerr and Ba˜nados-Teitelboim-Zanelli systems as a symmetry breaking phase transition, critical exponents are evalu- ated and compared with classical Landau predictions. We suggest a definition of isothermal compressibity which is independent of spin direction. We find identical exponents for both systems, and possible universality in the phase transitions of these systems. PACS numbers: 05.70.Fh, 04.20.Jb, 05.70.JK arXiv:gr-qc/0102075v1 16 Feb 2001 ∗Permanent address: Physics Department, University of Windsor, Ontario N9B 3P4, Canada 1 I. INTRODUCTION Black holes are perfect absorbers classically but do not emit anything; their classical physical temperature is absolute zero. In the semi-classical approximation black holes emit Hawking radiation with a perfect thermal spectrum. This allows a consistent interpretation of the laws of black hole mechanics as physically corresponding to the ordinary laws of thermodynamics. The classical laws of black hole mechanics together with the formula for the temperature of Hawking radiation allow one to identify the general relativistic horizon area, /4, as playing the mathematical role of entropy. The geometric thermodynamics of A black holes has provided many useful insights to the relation between geometric quantities and physics. These analogies are particularly useful when applied to constant mass black hole metric transformations involving a symmetry reduction. The Schwarzschild to Kerr (SCH-K) transform is an example.
    [Show full text]
  • Statistical Field Theory
    Statistical Field Theory R R Horgan [email protected] www.damtp.cam.ac.uk/user/rrh December 4, 2014 Contents 1 BOOKS iii 2 Introduction 1 3 Definitions, Notation and Statistical Physics 2 4 The D=1 Ising model and the transfer matrix 7 5 The Phenomenology of Phase Transitions 10 5.1 The general structure of phase diagrams . 16 5.1.1 Structures in a phase diagram: a description of Q . 16 6 Landau-Ginsburg theory and mean field theory 18 7 Mean field theory 22 8 The scaling hypothesis 25 9 Critical properties of the 1D Ising model 28 10 The blocking transformation 29 11 The Real Space Renormalization Group 35 12 The Partition Function and Field Theory 45 13 The Gaussian Model 47 i CONTENTS ii 14 The Perturbation Expansion 52 15 The Ginsberg Criterion for Dc 59 16 Calculation of the Critical Index 61 17 General Ideas 68 17.1 Domains and the Maxwell Construction . 69 17.2 Types of critical point . 69 17.3 Hysteresis . 72 17.4 Mean Field Theory . 73 1 BOOKS iii 1 BOOKS (1) \Statistical Mechanics of Phase Transitions" J.M. Yeomans, Oxford Scientific Publications. (2) \Statistical Physics", Landau and Lifshitz, Pergammon Press. (3) \The Theory of Critical Phenomena" J.J. Binney et al., Oxford Scientific Pub- lications. (4) \Scaling and Renormalization in Statistical Physics" John Cardy, Cambridge Lecture Notes in Physics. (5) \ Quantum and Statistical Field Theory" M. Le Bellac, Clarendon Press. (6) \Introduction to the Renormalisation Group of Critical Phenomena" P. Pfeuty and G. Toulouse, Wiley. ` (*7) \Fluctuation Theory of phase Transitions" Patashinskii and Pokrovskii, Perg- amon.
    [Show full text]
  • Black Hole Chemistry: Thermodynamics with Lambda Arxiv:1608.06147V3
    Prepared for submission to JHEP Black hole chemistry: thermodynamics with Lambda David Kubizˇn´ak1;2 Robert B. Mann2;1 Mae Teo3 1Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada 2Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada 3Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305, USA E-mail: [email protected], [email protected], [email protected] Abstract: We review recent developments on the thermodynamics of black holes in extended phase space, where the cosmological constant is interpreted as thermo- dynamic pressure and treated as a thermodynamic variable in its own right. In this approach, the mass of the black hole is no longer regarded as internal energy, rather it is identified with the chemical enthalpy. This leads to an extended dictionary for black hole thermodynamic quantities, in particular a notion of thermodynamic volume emerges for a given black hole spacetime. This volume is conjectured to satisfy the reverse isoperimetric inequality|an inequality imposing a bound on the amount of entropy black hole can carry for a fixed thermodynamic volume. New thermodynamic phase transitions naturally emerge from these identifications. Namely, we show that black holes can be understood from the viewpoint of chemistry, in terms of concepts arXiv:1608.06147v3 [hep-th] 6 May 2021 such as Van der Waals fluids, reentrant phase transitions, and triple points. We also re- view the recent attempts at extending the AdS/CFT dictionary in this setting, discuss the connections with horizon thermodynamics, applications to Lifshitz spacetimes, and outline possible future directions in this field.
    [Show full text]
  • Violating of the Classical Essam-Fisher and Rushbrooke Formulas for Quantum Phase Transitions V. Udodov Abstract the Critical E
    Violating of the classical Essam-Fisher and Rushbrooke formulas for quantum phase transitions V. Udodov Abstract The classical Essam-Fisher and Rushbrooke relationships (1963) that connect the equilibrium critical exponents of susceptibility, specific heat and order parameter are shown to be valid only if the critical temperature TС> 0 and T→TC. For quantum phase transitions (PT’s) with TC = 0K, these relations are proved to be of different form. This fact has been actually observed experimentally, but the reasons were not quite clear. A general formula containing the classical results as a special case is proposed. This formula is applicable to all equilibrium PT's of any space dimension. The predictions of the theory are consistent with the available experimental data and do not cast any doubts upon the scaling hypothesis. The critical exponents and scaling hypothesis (Widom (1965), Domb and Hunter (1965); Kadanoff (1966)) underlying the theory of critical phenomena [1- 3]. Essam and Fisher equality [4] and Rushbrooke inequality [5] (as the most famous inequality between critical exponents) were discovered 50 years ago and everyone believed that the relations are performed. These relations are connected with the critical exponents of the specific heat (α, α’), susceptibility (γ, γ’) and the order parameter β (determining of the critical exponents see below) [1-3]. In 1965- 66, it was shown that under the scaling hypothesis the Essam and Fisher equality are fulfilled and the Rushbrooke inequality '2 ' 2 reduces to equality [1] '2 ' 2. These results were obtained under the assumption that the critical temperature is finite TС > 0 and T→TC.
    [Show full text]
  • Violating of the Essam-Fisher and Rushbrooke Relationships at Low Temperatures
    World Journal of Condensed Matter Physics, 2015, 5, 55-59 Published Online May 2015 in SciRes. http://www.scirp.org/journal/wjcmp http://dx.doi.org/10.4236/wjcmp.2015.52008 Violating of the Essam-Fisher and Rushbrooke Relationships at Low Temperatures Vladimir Udodov Department of Theoretical Physics, Katanov Khakas State University, Abakan, Russia Email: [email protected] Received 4 January 2015; accepted 30 March 2015; published 1 April 2015 Copyright © 2015 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract The Essam-Fisher and Rushbrooke relationships (1963) that connect the equilibrium critical ex- ponents of susceptibility, specific heat and order parameter (and some other relations that follow from the scaling hypothesis) are shown to be valid only if the critical temperature TС > 0 and T → TC. For phase transitions (PT’s) with TC = 0 K these relations are proved to be of different form. This fact has been actually observed experimentally, but the reasons were not quite clear. A gen- eral formula containing the classical results as a special case is proposed. This formula is applica- ble to all equilibrium PT’s of any space dimension for both TC = 0 and TC > 0. The predictions of the theory are consistent with the available experimental data and do not cast any doubts upon the scaling hypothesis. Keywords Essam-Fisher and Rushbrooke Relationships, The Equilibrium Critical Exponents, The Scaling Hypothesis, Low Temperatures 1. Introduction The critical exponents and scaling hypothesis (Patashinsky and Pokrovsky (1964); Widom (1965); Domb and Hunter (1965); Kadanoff (1966)) underlying the theory of critical phenomena [1]-[5].
    [Show full text]
  • Revisiting the Scaling of the Specific Heat of the Three-Dimensional
    EPJ manuscript No. (will be inserted by the editor) Revisiting the scaling of the specific heat of the three-dimensional random-field Ising model Nikolaos G. Fytas1, Panagiotis E. Theodorakis2, and Alexander K. Hartmann3 1 Applied Mathematics Research Centre, Coventry University, Coventry CV1 5FB, United Kingdom 2 Institute of Physics, Polish Academy of Sciences, Al. Lotnik´ow 32/46, 02-668, Warsaw, Poland 3 Institut f¨ur Physik, Universit¨at Oldenburg, Carl-von-Ossietzky-Straße 9-11, 26111 Oldenburg, Germany Received: date / Revised version: date Abstract. We revisit the scaling behavior of the specific heat of the three-dimensional random-field Ising model with a Gaussian distribution of the disorder. Exact ground states of the model are obtained using graph-theoretical algorithms for different strengths h of the random fields and system sizes containing up to N = 2683 spins. By numerically differentiating the bond energy with respect to h, a specific-heat-like quantity is obtained whose maximum is found to converge to a constant at the thermodynamic limit. Compared to a previous study following the same approach, we have studied here much larger system sizes with an increased statistical accuracy. We discuss the relevance of our results under the prism of a modified Rushbrooke inequality for the case of a saturating specific heat. Finally, as a byproduct of our analysis, we provide high-accuracy estimates of the critical field hc = 2.279(7) and the critical exponent of the correlation exponent ν = 1.37(1), in excellent agreement to the most recent computations in the literature. PACS. PACS. 05.50+q Lattice theory and statistics (Ising, Potts.
    [Show full text]
  • On Entropy, Specific Heat, Susceptibility and Rushbrooke
    On entropy, specific heat, susceptibility and Rushbrooke inequality in percolation M. K. Hassan1, D. Alam2, Z. I. Jitu1 and M. M. Rahman1 1 Department of Physics, University of Dhaka, Dhaka 1000, Bangladesh 2 Department of Physics, University of Central Florida, Orlando, Florida 32816, USA (Dated: July 23, 2018) We investigate percolation, a probabilistic model for continuous phase transition (CPT), on square and weighted planar stochastic lattices. In its thermal counterpart, entropy is minimally low where order parameter (OP) is maximally high and vice versa. Besides, specific heat, OP and susceptibil- ity exhibit power-law when approaching the critical point and the corresponding critical exponents α,β,γ respectably obey the Rushbrooke inequality (RI) α + 2β + γ ≥ 2. Their analogues in per- colation, however, remain elusive. We define entropy, specific heat and redefine susceptibility for percolation and show that they behave exactly in the same way as their thermal counterpart. We also show that RI holds for both the lattices albeit they belong to different universality classes. PACS numbers: 61.43.Hv, 64.60.Ht, 68.03.Fg, 82.70.Dd The emergence of a well-defined critical value accom- arrive at a threshold value pc at which there appears a panied by a dramatic change in the order parameter (OP) spanning cluster for the first time that spans across the and entropy (S) without jump or discontinuity is an in- entire system. Such transition from isolated finite sized dication of second order or continuous phase transition clusters to spanning cluster across pc is found to be rem- (CPT) which has acquired a central focus in condensed iniscent of the CPT.
    [Show full text]
  • 81 Problems for Chapter 1 1.1 [Equivalence of Heat and Work]
    81 Problems for Chapter 1 1.1 [Equivalence of heat and work] A block of mass M = 1 g is at rest in space (vacuum). Another block of the same mass and velocity V = 1:5 km/s collides with the first block and the two blocks stick to each other. (1) Assuming that there is no radiation loss of energy and that there is no rotation of the resultant single block, obtain the temperature of the resultant single block after equilibration. Assume that the specific heat of the material is 2.1 J/g·K. (2) If rotation of the resultant body is allowed, what can be said about its final tem- perature? In particular, is it possible not to raise the temperature of the resultant single block? (Only a qualitative discussion will be enough.) 1.2 [Exchange of temperatures] Suppose there are two water tanks A and B containing the same amount of water. Initially, A is 42◦C and B is 25◦C. The final state we want is A to be 25◦C and B 42◦C (that is, the temperatures of A and B are exchanged; e.g., A is the used warm bath water, and B new clean tap water). Assume that the whole system is thermally isolated. (1) Is the process reversibly feasible? Explain why. (2) Devise a process. No precise statement is needed; only state key ideas. (3) If the reader's process in (2) contains mechanical parts (e.g., a heat engine), de- vise a process without any moving parts. No precise statement is needed; only state key ideas.
    [Show full text]
  • Thermodynamics of Hairy Black Holes in Lovelock Gravity
    Prepared for submission to JHEP Thermodynamics of hairy black holes in Lovelock gravity Robie A. Hennigar,a Erickson Tjoab;a and Robert B. Manna aDepartment of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1 bDivision of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore, 637371 E-mail: [email protected], [email protected], [email protected] Abstract: We perform a thorough study of the thermodynamic properties of a class of Lovelock black holes with conformal scalar hair arising from coupling of a real scalar field to the dimensionally extended Euler densities. We study the linearized equations of motion of the theory and describe constraints under which the theory is free from ghosts/tachyons. We then consider, within the context of black hole chemistry, the thermodynamics of the hairy black holes in the Gauss-Bonnet and cubic Lovelock theories. We clarify the con- nection between isolated critical points and thermodynamic singularities, finding a one parameter family of these critical points which occur for well-defined thermodynamic pa- rameters. We also report on a number of novel results, including `virtual triple points' and the first example of a `λ-line'|a line of second order phase transitions|in black hole thermodynamics. arXiv:1612.06852v3 [hep-th] 21 Feb 2017 Contents 1 Introduction1 2 Exact Solution & Thermodynamics3 3 Equations of motion and ghosts7 3.1 Einstein case8 3.2 Gauss-Bonnet and Cubic Lovelock
    [Show full text]