Phys Midterm. March 17

Save this PDF as:

Size: px
Start display at page:

Transcription

1 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 magnetic moment, and s takes values 1 and 1 (a Calculate the partition function of this system Z = s=1, 1 ( e E s µh = e + e µh µh = 2 cosh (b Calculate the Helmholtz free energy F, entropy S, and heat capacity c Calculate also energy E as a function of temperature [ ( µh F = log Z = log 2 cosh ] S = F [ ( µh = log 2 cosh c = = ] µh ( µh tanh 2 µh cosh ( µh At small the specific heat decays exponentially, ( 2µH c 2 e 2µH, µh At large the specific heat decays as a power law ( µh c 2, µh

2 Finally, energy E is given by ( µh E = F + S = µh tanh If the temperature is taken to zero, the energy goes to µh, that is, the spin is in its ground state If the temperature is taken to infinity, the energy goes to zero his is due to spin spending equal time in ground and excited states, which are exactly opposite to each other in value So the average energy is then equal to zero Notice that the heat capacity can be alternatively calculated as c = E (c Average magnetization M is defined as in M = µ where p s is the probability for the variable s to take values 1 and 1 Show that once the free energy F is known, one can compute the average magnetization by using M = F H s p s, Use this relation to find the average magnetization M Magnetic susceptibility χ is defined in the following way χ = M H H=0 Calculate magnetic susceptibility for this problem What happens to the magnetic susceptibility as temperature is taken to 0? p s is the probability that the energy of the spin is E s In the Gibbs distribution, it takes values Es p s = e Z hen the average magnetization is given by On the other hand, M = µ s e µhs Z F = log Z = log e µhs 2

3 Differentiating this expression with respect to H we find F H = µ Z s=1, 1 s e µh, which coincides with the definition of M Now we calculate M M = F ( µh H = µ tanh Finally, we calculate the magnetic susceptibility χ = M = µ2 H H=0 Magnetic susceptibility diverges as 1/ as temperature is taken to zero (d Calculate F, S, c, and χ for the system which consists of N spins F is an extensive quantity, thus F N = NF, where F N is the free energy of N spins, and F is the free energy of one spin calculated above Since S, c, and χ are all derived by differentiating F, S N, c N, and χ N will all be N times the appropriate quantities for one spin More detailed calculate takes into account that the energy of the system consists of sums of energies of each spin, and each of them is summed over independently in the partition function So the partition function for N spins is Z N = Z N, where Z is the partition function for one spin hus, F N = NF, and so on (e Suppose this spin system is put into a state where its temperature is negative < 0 (this is, in fact, possible and has been achieved experimentally with nuclear spins hen it is brought into a contact with a large heat reservoir which is at room temperature reservoir +300K Will the energy of the spin system be increasing or decreasing, and how will the temperature of the spin system be changing in time, until it reaches equilibrium with the reservoir? Use the law of entropy increase and apply it to the combined spin+reservoir system to answer this question Negative temperatures are possible in the systems whose spectrum is bounded from above If the temperature is negative and large, < 0 and µh, the results of the end of section (b where energy is calculated imply that the energy becomes E = µh In other words, at large negative temperature the system spends most of its time in its 3

4 excited state his can be contrasted with the situation at infinite temperature where the system divides its time equally between all states, and the zero temperature situation where the system is in its ground state his in fact means that the negative temperatures should be thought of as not being below positive temperatures, but rather being above infinite temperature As the temperature of the system is increased, its energy increases too, and reaches certain value as the temperature becomes infinite o increase its energy further, one needs to make the temperature negative infinite, and then continue to increase it until it hits zero from below, which is, in effect, a maximum possible temperature his could be expressed mathematically by introducing a notation β = 1 his parameter changes from to 0, as temperature is taken from 0 to As temperature is taken from to 0, β changes from 0 to hen the energy of the spin can be written as E = µh tanh (βµh, which is a monotonously increasing function of β Already these considerations lead to the conclusion that a system at negative temperature is hotter than a system at positive temperature hus if it is brought in contact with a reservoir at positive temperature, its temperature will decrease, go through, then, and then continue to decrease further until it reaches the temperature of the reservoir he same conclusion can be reached via analysis of the entropy he total entropy of the combined system is given by S(E = S(E + S reservoir (E tot E E = E reservoir(e tot E E tot = 1 1 reservoir < 0 since < 0, reservoir > 0 hus if E increases, total entropy will decrease Since the total entropy should always increase (law of entropy increase, the energy E will be decreasing In other words, the energy will flow from the system at negative temperature towards the reservoir at positive temperature 4

5 Useful formulae Canonical ensemble Partition function Z = n e E n, where E n are the energy levels of the system Free energy F = log Z Energy of the system E = F + S df = Sd P dv + µdn, de = ds P dv + µdn Probability that the system s energy is equal to E n is p n = 1 Z e E n Grand canonical ensemble Partition function Z = n e E n + µnn, where E n are the energy levels of the system, and N n is the number of particles at this level Grand canonical free energy Ω = log Z dω = ds P dv Ndµ Ω = F µn Probability that the system s energy is equal to E n if it has N n particles is p n = 1 E Z e n + µn n Other formulae Heat capacity is the amount of heat pumped into the system to increase its temperature by d At fixed volume it is given by c v = ( At fixed pressure it is V given by c p = ( Specific heat is the heat capacity per particle P Maxwell relations are relations between derivatives which follow from the fact that partial second derivatives of thermodynamic potentials do not depend on the order of differentiation For example, ( ( P de = ds P dv = V S V Quasiclassical approximation to the partition function of a particle d 3 xd 3 p p Z = 2 (2π h 3 e 2m Equipartition theorem: specific heat of a quasiclassical system is equal to 1/2 per degree of freedom which enters its energy quadratically Fermi-Dirac and Bose-Einstein grand canonical free energy Ω = s n ln ( 1 + s e E n + µ, where s = 1 for Fermi-Dirac and s = 1 for Bose-Einstein distributions Bose-Einstein distribution N = N 0 + d 3 p d 3 x (2π h 3 1 e p2 2m µ 1, where N is the total number of particles, and N 0 are the particles in the condensate N 0 = 0 at > c, while µ = 0 at < c 5

Part II: Statistical Physics

Chapter 6: Boltzmann Statistics SDSMT, Physics Fall Semester: Oct. - Dec., 2014 1 Introduction: Very brief 2 Boltzmann Factor Isolated System and System of Interest Boltzmann Factor The Partition Function

Thermal and Statistical Physics Department Exam Last updated November 4, L π

Thermal and Statistical Physics Department Exam Last updated November 4, 013 1. a. Define the chemical potential µ. Show that two systems are in diffusive equilibrium if µ 1 =µ. You may start with F =

Physics 408 Final Exam

Physics 408 Final Exam Name You are graded on your work (with partial credit where it is deserved) so please do not just write down answers with no explanation (or skip important steps)! Please give clear,

i=1 n i, the canonical probabilities of the micro-states [ βǫ i=1 e βǫn 1 n 1 =0 +Nk B T Nǫ 1 + e ǫ/(k BT), (IV.75) E = F + TS =

IV.G Examples The two examples of sections (IV.C and (IV.D are now reexamined in the canonical ensemble. 1. Two level systems: The impurities are described by a macro-state M (T,. Subject to the Hamiltonian

Lecture Notes, Statistical Mechanics (Theory F) Jörg Schmalian

Lecture Notes, Statistical Mechanics Theory F) Jörg Schmalian April 14, 2014 2 Institute for Theory of Condensed Matter TKM) Karlsruhe Institute of Technology Summer Semester, 2014 Contents 1 Introduction

Landau s Fermi Liquid Theory

Thors Hans Hansson Stockholm University Outline 1 Fermi Liquids Why, What, and How? Why Fermi liquids? What is a Fermi liquids? Fermi Liquids How? 2 Landau s Phenomenological Approach The free Fermi gas

2. Under conditions of constant pressure and entropy, what thermodynamic state function reaches an extremum? i

1. (20 oints) For each statement or question in the left column, find the appropriate response in the right column and place the letter of the response in the blank line provided in the left column. 1.

Chapter 3. Entropy, temperature, and the microcanonical partition function: how to calculate results with statistical mechanics.

Chapter 3. Entropy, temperature, and the microcanonical partition function: how to calculate results with statistical mechanics. The goal of equilibrium statistical mechanics is to calculate the density

Physics 607 Final Exam

Physics 607 Final Exam Please show all significant steps clearly in all problems. 1. Let E,S,V,T, and P be the internal energy, entropy, volume, temperature, and pressure of a system in thermodynamic equilibrium

Statistical Mechanics Notes. Ryan D. Reece

Statistical Mechanics Notes Ryan D. Reece August 11, 2006 Contents 1 Thermodynamics 3 1.1 State Variables.......................... 3 1.2 Inexact Differentials....................... 5 1.3 Work and Heat..........................

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Statistical Physics I Spring Term 2013 Notes on the Microcanonical Ensemble

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.044 Statistical Physics I Spring Term 2013 Notes on the Microcanonical Ensemble The object of this endeavor is to impose a simple probability

Quantum Grand Canonical Ensemble

Chapter 16 Quantum Grand Canonical Ensemble How do we proceed quantum mechanically? For fermions the wavefunction is antisymmetric. An N particle basis function can be constructed in terms of single-particle

IV. Classical Statistical Mechanics

IV. Classical Statistical Mechanics IV.A General Definitions Statistical Mechanics is a probabilistic approach to equilibrium macroscopic properties of large numbers of degrees of freedom. As discussed

Solution to the exam in TFY4230 STATISTICAL PHYSICS Friday december 19, 2014

NTNU Page 1 of 10 Institutt for fysikk Fakultet for fysikk, informatikk og matematikk This solution consists of 10 pages. Problem 1. Cold Fermi gases Solution to the exam in TFY4230 STATISTICAL PHYSICS

PHYS First Midterm SOLUTIONS

PHYS 430 - First Midterm SOLUIONS 1. Comment on the following concepts (Just writing equations will not gain you any points): (3 points each, 21 points total) (a) Negative emperatures emperature is a measure

Chapter 14. Ideal Bose gas Equation of state

Chapter 14 Ideal Bose gas In this chapter, we shall study the thermodynamic properties of a gas of non-interacting bosons. We will show that the symmetrization of the wavefunction due to the indistinguishability

Set 3: Thermal Physics

Set 3: Thermal Physics Equilibrium Thermal physics describes the equilibrium distribution of particles for a medium at temperature T Expect that the typical energy of a particle by equipartition is E kt,

Energy Barriers and Rates - Transition State Theory for Physicists

Energy Barriers and Rates - Transition State Theory for Physicists Daniel C. Elton October 12, 2013 Useful relations 1 cal = 4.184 J 1 kcal mole 1 = 0.0434 ev per particle 1 kj mole 1 = 0.0104 ev per particle

Statistical Thermodynamics - Fall Professor Dmitry Garanin. Thermodynamics. February 15, 2017 I. PREFACE

1 Statistical hermodynamics - Fall 2009 rofessor Dmitry Garanin hermodynamics February 15, 2017 I. REFACE he course of Statistical hermodynamics consist of two parts: hermodynamics and Statistical hysics.

Statistical Physics a second course

Statistical Physics a second course Finn Ravndal and Eirik Grude Flekkøy Department of Physics University of Oslo September 3, 2008 2 Contents 1 Summary of Thermodynamics 5 1.1 Equations of state..........................

Chapter 6. Phase transitions. 6.1 Concept of phase

Chapter 6 hase transitions 6.1 Concept of phase hases are states of matter characterized by distinct macroscopic properties. ypical phases we will discuss in this chapter are liquid, solid and gas. Other

I.G Approach to Equilibrium and Thermodynamic Potentials

I.G Approach to Equilibrium and Thermodynamic otentials Evolution of non-equilibrium systems towards equilibrium is governed by the second law of thermodynamics. For eample, in the previous section we

[S R (U 0 ɛ 1 ) S R (U 0 ɛ 2 ]. (0.1) k B

Canonical ensemble (Two derivations) Determine the probability that a system S in contact with a reservoir 1 R to be in one particular microstate s with energy ɛ s. (If there is degeneracy we are picking

UNIVERSITY COLLEGE LONDON. University of London EXAMINATION FOR INTERNAL STUDENTS. For The Following Qualifications:-

UNIVERSITY COLLEGE LONDON University of London EXAMINATION FOR INTERNAL STUDENTS For The Following Qualifications:- B. Sc. M. Sci. Physics 2B28: Statistical Thermodynamics and Condensed Matter Physics

Benchmarking the Hartree-Fock and Hartree-Fock-Bogoliubov approximations to level densities. G.F. Bertsch, Y. Alhassid, C.N. Gilbreth, and H.

Benchmarking the Hartree-Fock and Hartree-Fock-Bogoliubov approximations to level densities G.F. Bertsch, Y. Alhassid, C.N. Gilbreth, and H. Nakada 5th Workshop on Nuclear Level Density and Gamma Strength,

PC5202 Advanced Statistical Mechanics. Midterm test, closed book, 1 March 2017

PC50 Advanced Statistical Mechanics Midterm test, closed book, 1 March 017 1. Answer briefly the short questions below. Explain your symbols. a. What is the Boltzmann principle? b. Give a quick derivation

Grand-canonical ensembles

Grand-canonical ensembles As we know, we are at the point where we can deal with almost any classical problem (see below), but for quantum systems we still cannot deal with problems where the translational

2m + U( q i), (IV.26) i=1

I.D The Ideal Gas As discussed in chapter II, micro-states of a gas of N particles correspond to points { p i, q i }, in the 6N-dimensional phase space. Ignoring the potential energy of interactions, the

Chapter 3. Property Relations The essence of macroscopic thermodynamics Dependence of U, H, S, G, and F on T, P, V, etc.

Chapter 3 Property Relations The essence of macroscopic thermodynamics Dependence of U, H, S, G, and F on T, P, V, etc. Concepts Energy functions F and G Chemical potential, µ Partial Molar properties

to satisfy the large number approximations, W W sys can be small.

Chapter 12. The canonical ensemble To discuss systems at constant T, we need to embed them with a diathermal wall in a heat bath. Note that only the system and bath need to be large for W tot and W bath

Metropolis 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

TSTC Lectures: Theoretical & Computational Chemistry

TSTC Lectures: Theoretical & Computational Chemistry Rigoberto Hernandez May 2009, Lecture #2 Statistical Mechanics: Structure S = k B log W! Boltzmann's headstone at the Vienna Zentralfriedhof.!! Photo

Last Name: First Name ID

Last Name: First Name ID This is a set of practice problems for the final exam. It is not meant to represent every topic and is not meant to be equivalent to a 2-hour exam. These problems have not been

Internal Degrees of Freedom

Physics 301 16-Oct-2002 15-1 Internal Degrees of Freedom There are several corrections we might make to our treatment of the ideal gas If we go to high occupancies our treatment using the Maxwell-Boltzmann

...Thermodynamics. Entropy: The state function for the Second Law. Entropy ds = d Q. Central Equation du = TdS PdV

...Thermodynamics Entropy: The state function for the Second Law Entropy ds = d Q T Central Equation du = TdS PdV Ideal gas entropy s = c v ln T /T 0 + R ln v/v 0 Boltzmann entropy S = klogw Statistical

Aus: Daniel Schroeder An introduction to thermal physics, Pearson Verlag

Ω A q B Ω B Ω total 0 00 8 0 8 8 0 8 00 99 9 0 80 8 0 8 0 98 0 80 0 8 00 97 0 0 80 0 8 0 8 9 0 79 0 88 9 0 8 0 8 0 0 0 9 0 0 9 0 77 0 9 9 88 0 8 0 00 7 0 9 0 7 0 9 9 0 Ω total ( 0 ) 7 0 0 0 00 N, q few

Thermodynamics of nuclei in thermal contact

Thermodynamics of nuclei in thermal contact Karl-Heinz Schmidt, Beatriz Jurado CENBG, CNRS/IN2P3, Chemin du Solarium B.P. 120, 33175 Gradignan, France Abstract: The behaviour of a di-nuclear system in

Definite Integral and the Gibbs Paradox

Acta Polytechnica Hungarica ol. 8, No. 4, 0 Definite Integral and the Gibbs Paradox TianZhi Shi College of Physics, Electronics and Electrical Engineering, HuaiYin Normal University, HuaiAn, JiangSu, China,

1 Ensemble of three-level systems

PHYS 607: Statistical Physics. Fall 2015. Home Assignment 2 Entropy, Energy, Heat Capacity Katrina Colletti September 23, 2015 1 Ensemble of three-level systems We have an ensemble of N atoms, with N 1,

ChE 503 A. Z. Panagiotopoulos 1

ChE 503 A. Z. Panagiotopoulos 1 STATISTICAL MECHANICAL ENSEMLES 1 MICROSCOPIC AND MACROSCOPIC ARIALES The central question in Statistical Mechanics can be phrased as follows: If particles (atoms, molecules,

Preliminary Examination - Day 2 May 16, 2014

UNL - Department of Physics and Astronomy Preliminary Examination - Day May 6, 04 This test covers the topics of Thermodynamics and Statistical Mechanics (Topic ) and Mechanics (Topic ) Each topic has

Equilibrium Statistical Mechanics

Equilibrium Statistical Mechanics October 14, 2007 Chapter I What is Statistical Mechanics? Microscopic and macroscopic descriptions. Statistical mechanics is used to describe systems with a large ( 10

Insigh,ul Example. E i = n i E, n i =0, 1, 2,..., 8. N(n 0,n 1,n 2,..., n 8 )= n 1!n 2!...n 8!

STATISTICS Often the number of particles in a system is prohibitively large to solve detailed state (from Schrodinger equation) or predict exact motion (using Newton s laws). Bulk properties (pressure,

!W "!#U + T#S, where. !U = U f " U i and!s = S f " S i. ! W. The maximum amount of work is therefore given by =!"U + T"S. !W max

1 Appendix 6: The maximum wk theem (Hiroshi Matsuoka) 1. Question and answer: the maximum amount of wk done by a system Suppose we are given two equilibrium states of a macroscopic system. Consider all

The answer (except for Z = 19) can be seen in Figure 1. All credit to McGraw- Hill for the image.

Problem 9.3 Which configuration has a greater number of unpaired spins? Which one has a lower energy? [Kr]4d 9 5s or [Kr]4d. What is the element and how does Hund s rule apply?. Solution The element is

The goal of equilibrium statistical mechanics is to calculate the diagonal elements of ˆρ eq so we can evaluate average observables < A >= Tr{Â ˆρ eq

Chapter. The microcanonical ensemble The goal of equilibrium statistical mechanics is to calculate the diagonal elements of ˆρ eq so we can evaluate average observables < A >= Tr{Â ˆρ eq } = A that give

Joint Entrance Examination for Postgraduate Courses in Physics EUF

Joint Entrance Examination for Postgraduate Courses in Physics EUF Second Semester 013 Part 1 3 April 013 Instructions: DO NOT WRITE YOUR NAME ON THE TEST. It should be identified only by your candidate

Statistical Mechanics. Henri J.F. Jansen Department of Physics Oregon State University

Statistical Mechanics Henri J.F. Jansen Department of Physics Oregon State University October 2, 2008 II Contents Foundation of statistical mechanics.. Introduction..............................2 Program

8.333: Statistical Mechanics I Re: 2007 Final Exam. Review Problems

8.333: Statistical Mechanics I Re: 007 Final Exam Review Problems The enclosed exams (and solutions) from the previous years are intended to help you review the material. ******** Note that the first parts

LECTURE NOTES ON STATISTICAL MECHANICS. Scott Pratt Department of Physics and Astronomy Michigan State University PHY

LECTURE NOTES ON STATISTICAL MECHANICS Scott Pratt Department of Physics and Astronomy Michigan State University PHY 831 2007-2016 FORWARD These notes are for the one-semester graduate level statistical

INTRODUCTION TO STATISTICAL MECHANICS Exercises

École d hiver de Probabilités Semaine 1 : Mécanique statistique de l équilibre CIRM - Marseille 4-8 Février 2012 INTRODUCTION TO STATISTICAL MECHANICS Exercises Course : Yvan Velenik, Exercises : Loren

Phase structure of balck branes

Phase structure of balck branes Zhiguang Xiao Collaborators: J. X. Lu, Shibaji Roy December 2, 2010 Introduction Canonical ensemble Grand canonical ensemble Summary and Disscusion Introduction: Partition

PY2005: Thermodynamics

ome Multivariate Calculus Y2005: hermodynamics Notes by Chris Blair hese notes cover the enior Freshman course given by Dr. Graham Cross in Michaelmas erm 2007, except for lecture 12 on phase changes.

Information to energy conversion in an electronic Maxwell s demon and thermodynamics of measurements.

Information to energy conversion in an electronic Maxwell s demon and thermodynamics of measurements Stony Brook University, SUNY Dmitri V Averin and iang Deng Low-Temperature Lab, Aalto University Jukka

UNIVERSITY COLLEGE LONDON. University of London EXAMINATION FOR INTERNAL STUDENTS. For The Following Qualifications:-

UNIVERSITY COLLEGE LONDON University of London EXAMINATION FOR INTERNAL STUDENTS For The Following Qualifications:- B.Sc. M.Sci. Statistical Thermodynamics COURSE CODE : PHAS2228 UNIT VALUE : 0.50 DATE

Spontaneous 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

Application of Mean-Field Jordan Wigner Transformation to Antiferromagnet System

Commun. Theor. Phys. Beijing, China 50 008 pp. 43 47 c Chinese Physical Society Vol. 50, o. 1, July 15, 008 Application of Mean-Field Jordan Wigner Transformation to Antiferromagnet System LI Jia-Liang,

Thermodynamics: A Brief Introduction. Thermodynamics: A Brief Introduction

Brief review or introduction to Classical Thermodynamics Hopefully you remember this equation from chemistry. The Gibbs Free Energy (G) as related to enthalpy (H) and entropy (S) and temperature (T). Δ

Chapter 10: Statistical Thermodynamics

Chapter 10: Statistical Thermodynamics Chapter 10: Statistical Thermodynamics...125 10.1 The Statistics of Equilibrium States...125 10.2 Liouville's Theorem...127 10.3 The Equilibrium Distribution Function...130

Introduction to Statistical Physics

Introduction to Statistical Physics Rigorous and comprehensive, this textbook introduces undergraduate students to simulation methods in statistical physics. The book covers a number of topics, including

Caltech Ph106 Fall 2001

Caltech h106 Fall 2001 ath for physicists: differential forms Disclaimer: this is a first draft, so a few signs might be off. 1 Basic properties Differential forms come up in various parts of theoretical

Physics 622. T.R. Lemberger. Jan. 2003

Physics 622. T.R. Lemberger. Jan. 2003 Connections between thermodynamic quantities: enthalpy, entropy, energy, and microscopic quantities: kinetic and bonding energies. Useful formulas for one mole of

The canonical ensemble

The canonical ensemble Anders Malthe-Sørenssen 23. september 203 Boltzmann statistics We have so far studied systems with constant N, V, and E systems in the microcanonical ensemble. Such systems are often

P340: Thermodynamics and Statistical Physics, Exam#2 Any answer without showing your work will be counted as zero

P340: hermodynamics and Statistical Physics, Exam#2 Any answer without showing your work will be counted as zero 1. (15 points) he equation of state for the an der Waals gas (n = 1 mole) is (a) Find (

Chemistry. Lecture 10 Maxwell Relations. NC State University

Chemistry Lecture 10 Maxwell Relations NC State University Thermodynamic state functions expressed in differential form We have seen that the internal energy is conserved and depends on mechanical (dw)

ADSORPTION IN MICROPOROUS MATERIALS: ANALYTICAL EQUATIONS FOR TYPE I ISOTHERMS AT HIGH PRESSURE

ADSORPTION IN MICROPOROUS MATERIALS: ANALYTICAL EQUATIONS FOR TYPE I ISOTHERMS AT HIGH PRESSURE A. L. MYERS Department of Chemical and Biomolecular Engineering University of Pennsylvania, Philadelphia

+ 1. which gives the expected number of Fermions in energy state ɛ. The expected number of Fermions in energy range ɛ to ɛ + dɛ is then dn = n s g s

Chapter 8 Fermi Systems 8.1 The Perfect Fermi Gas In this chapter, we study a gas of non-interacting, elementary Fermi particles. Since the particles are non-interacting, the potential energy is zero,

Section 4 Statistical Thermodynamics of delocalised particles

Section 4 Statistical Thermodynamics of delocalised particles 4.1 Classical Ideal Gas 4.1.1 Indistinguishability Since the partition function is proportional to probabilities it follows that for composite

4.1 Constant (T, V, n) Experiments: The Helmholtz Free Energy

Chapter 4 Free Energies The second law allows us to determine the spontaneous direction of of a process with constant (E, V, n). Of course, there are many processes for which we cannot control (E, V, n)

arxiv:cond-mat/ v1 [cond-mat.stat-mech] 25 Jul 2003

APS/123-QED Statistical Mechanics in the Extended Gaussian Ensemble Ramandeep S. Johal, Antoni Planes, and Eduard Vives Departament d Estructura i Constituents de la Matèria, Universitat de Barcelona Diagonal

Unit 7 (B) Solid state Physics

Unit 7 (B) Solid state Physics hermal Properties of solids: Zeroth law of hermodynamics: If two bodies A and B are each separated in thermal equilibrium with the third body C, then A and B are also in

Cold Metastable Neon Atoms Towards Degenerated Ne*- Ensembles

Cold Metastable Neon Atoms Towards Degenerated Ne*- Ensembles Supported by the DFG Schwerpunktprogramm SPP 1116 and the European Research Training Network Cold Quantum Gases Peter Spoden, Martin Zinner,

Phase Transitions in Physics and Computer Science. Cristopher Moore University of New Mexico and the Santa Fe Institute

Phase Transitions in Physics and Computer Science Cristopher Moore University of New Mexico and the Santa Fe Institute Magnetism When cold enough, Iron will stay magnetized, and even magnetize spontaneously

Micro-canonical ensemble model of particles obeying Bose-Einstein and Fermi-Dirac statistics

Indian Journal o Pure & Applied Physics Vol. 4, October 004, pp. 749-757 Micro-canonical ensemble model o particles obeying Bose-Einstein and Fermi-Dirac statistics Y K Ayodo, K M Khanna & T W Sakwa Department

Lecture 25 Goals: Chapter 18 Understand the molecular basis for pressure and the idealgas

Lecture 5 Goals: Chapter 18 Understand the molecular basis for pressure and the idealgas law. redict the molar specific heats of gases and solids. Understand how heat is transferred via molecular collisions

Derivation of the Boltzmann Distribution

CLASSICAL CONCEPT REVIEW 7 Derivation of the Boltzmann Distribution Consider an isolated system, whose total energy is therefore constant, consisting of an ensemble of identical particles 1 that can exchange

Solutions to Problem Set 3

Cornell University, Physics Department Fall 014 PHYS-3341 Statistical Physics Prof. Itai Cohen Solutions to Problem Set 3 David C. Tsang, Woosong Choi 3.1 Boundary Conditions (a) We have the energy of

The fine-grained Gibbs entropy

Chapter 12 The fine-grained Gibbs entropy 12.1 Introduction and definition The standard counterpart of thermodynamic entropy within Gibbsian SM is the socalled fine-grained entropy, or Gibbs entropy. It

The 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

Spin- and heat pumps from approximately integrable spin-chains Achim Rosch, Cologne

Spin- and heat pumps from approximately integrable spin-chains Achim Rosch, Cologne Zala Lenarčič, Florian Lange, Achim Rosch University of Cologne theory of weakly driven quantum system role of approximate

arxiv: v2 [nucl-th] 15 Jun 2017

Kinetic freeze-out temperatures in central and peripheral collisions: Which one is larger? Hai-Ling Lao, Fu-Hu Liu, Bao-Chun Li, Mai-Ying Duan Institute of heoretical Physics, Shanxi University, aiyuan,

Homework Week 8 G = H T S. Given that G = H T S, using the first and second laws we can write,

Statistical Molecular hermodynamics University of Minnesota Homework Week 8 1. By comparing the formal derivative of G with the derivative obtained taking account of the first and second laws, use Maxwell

Quantum Electronics/Laser Physics Chapter 4 Line Shapes and Line Widths

Quantum Electronics/Laser Physics Chapter 4 Line Shapes and Line Widths 4.1 The Natural Line Shape 4.2 Collisional Broadening 4.3 Doppler Broadening 4.4 Einstein Treatment of Stimulated Processes Width

CHAPTER 1. Thermostatics

CHAPTER 1 Thermostatics 1. Context and origins The object of thermodynamics is to describe macroscopic systems (systems that are extended in space) that vary slowly in time. Macroscopic systems are formed

PHYSICS DEPARTMENT, PRINCETON UNIVERSITY PHYSICS 301 MIDTERM EXAMINATION. October 22, 2003, 10:00 10:50 am, Jadwin A06 SOLUTIONS

PHYSICS DEPARTMENT, PRINCETON UNIVERSITY PHYSICS 31 MIDTERM EXAMINATION October 22, 23, 1: 1:5 am, Jadwin A6 SOUTIONS This exam contains two problems. Work both problems. The problems count equally although

8.333: Statistical Mechanics I Problem Set # 5 Due: 11/22/13 Interacting particles & Quantum ensembles

8.333: Statistical Mechanics I Problem Set # 5 Due: 11/22/13 Interacting particles & Quantum ensembles 1. Surfactant condensation: N surfactant molecules are added to the surface of water over an area

Linear Response and Onsager Reciprocal Relations

Linear Response and Onsager Reciprocal Relations Amir Bar January 1, 013 Based on Kittel, Elementary statistical physics, chapters 33-34; Kubo,Toda and Hashitsume, Statistical Physics II, chapter 1; and

Phase transition. Asaf Pe er Background

Phase transition Asaf Pe er 1 November 18, 2013 1. Background A hase is a region of sace, throughout which all hysical roerties (density, magnetization, etc.) of a material (or thermodynamic system) are

Contents 1 Principles of Thermodynamics 2 Thermodynamic potentials 3 Applications of thermodynamics

Contents 1 Principles of Thermodynamics 1 1.1 Introduction............................. 1 1. State variables and differential forms.............. 1.3 Equation of state.......................... 4 1.4 Zeroth

Statistical. mechanics

CHAPTER 15 Statistical Thermodynamics 1: The Concepts I. Introduction. A. Statistical mechanics is the bridge between microscopic and macroscopic world descriptions of nature. Statistical mechanics macroscopic

although Boltzmann used W instead of Ω for the number of available states.

Lecture #13 1 Lecture 13 Obectives: 1. Ensembles: Be able to list the characteristics of the following: (a) icrocanonical (b) Canonical (c) Grand Canonical 2. Be able to use Lagrange s method of undetermined

Ensemble equivalence for non-extensive thermostatistics

Physica A 305 (2002) 52 57 www.elsevier.com/locate/physa Ensemble equivalence for non-extensive thermostatistics Raul Toral a;, Rafael Salazar b a Instituto Mediterraneo de Estudios Avanzados (IMEDEA),

The Black Body Radiation = Chapter 4 of Kittel and Kroemer The Planck distribution Derivation Black Body Radiation Cosmic Microwave Background The genius of Max Planck Other derivations Stefan Boltzmann

Spontaneous Symmetry Breaking in Bose-Einstein Condensates

The 10th US-Japan Joint Seminar Spontaneous Symmetry Breaking in Bose-Einstein Condensates Masahito UEDA Tokyo Institute of Technology, ERATO, JST collaborators Yuki Kawaguchi (Tokyo Institute of Technology)

Bose Einstein condensation of magnons and spin wave interactions in quantum antiferromagnets

Bose Einstein condensation of magnons and spin wave interactions in quantum antiferromagnets Talk at Rutherford Appleton Lab, March 13, 2007 Peter Kopietz, Universität Frankfurt collaborators: Nils Hasselmann,

Quantum Reservoir Engineering

Departments of Physics and Applied Physics, Yale University Quantum Reservoir Engineering Towards Quantum Simulators with Superconducting Qubits SMG Claudia De Grandi (Yale University) Siddiqi Group (Berkeley)

Statistical mechanics in the extended Gaussian ensemble

PHYSICAL REVIEW E 68, 563 23 Statistical mechanics in the extended Gaussian ensemble Ramandeep S. Johal,* Antoni Planes, and Eduard Vives Departament d Estructura i Constituents de la atèria, Facultat

THE CANONICAL ENSEMBLE 4.2 THE UNIFORM QUANTUM ENSEMBLE 64

THE CANONICAL ENSEMBLE 4.2 THE UNIFORM QUANTUM ENSEMBLE 64 Chapter 4 The Canonical Ensemble 4.1 QUANTUM ENSEMBLES A statistical ensemble for a dynamical system is a collection of system points in the system