Physics 451 - Statistical Mechanics II - Course Notes
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Canonical Ensemble
ME346A Introduction to Statistical Mechanics { Wei Cai { Stanford University { Win 2011 Handout 8. Canonical Ensemble January 26, 2011 Contents Outline • In this chapter, we will establish the equilibrium statistical distribution for systems maintained at a constant temperature T , through thermal contact with a heat bath. • The resulting distribution is also called Boltzmann's distribution. • The canonical distribution also leads to definition of the partition function and an expression for Helmholtz free energy, analogous to Boltzmann's Entropy formula. • We will study energy fluctuation at constant temperature, and witness another fluctuation- dissipation theorem (FDT) and finally establish the equivalence of micro canonical ensemble and canonical ensemble in the thermodynamic limit. (We first met a mani- festation of FDT in diffusion as Einstein's relation.) Reading Assignment: Reif x6.1-6.7, x6.10 1 1 Temperature For an isolated system, with fixed N { number of particles, V { volume, E { total energy, it is most conveniently described by the microcanonical (NVE) ensemble, which is a uniform distribution between two constant energy surfaces. const E ≤ H(fq g; fp g) ≤ E + ∆E ρ (fq g; fp g) = i i (1) mc i i 0 otherwise Statistical mechanics also provides the expression for entropy S(N; V; E) = kB ln Ω. In thermodynamics, S(N; V; E) can be transformed to a more convenient form (by Legendre transform) of Helmholtz free energy A(N; V; T ), which correspond to a system with constant N; V and temperature T . Q: Does the transformation from N; V; E to N; V; T have a meaning in statistical mechanics? A: The ensemble of systems all at constant N; V; T is called the canonical NVT ensemble. -
Bose-Einstein Condensation of Photons and Grand-Canonical Condensate fluctuations
Bose-Einstein condensation of photons and grand-canonical condensate fluctuations Jan Klaers Institute for Applied Physics, University of Bonn, Germany Present address: Institute for Quantum Electronics, ETH Zürich, Switzerland Martin Weitz Institute for Applied Physics, University of Bonn, Germany Abstract We review recent experiments on the Bose-Einstein condensation of photons in a dye-filled optical microresonator. The most well-known example of a photon gas, pho- tons in blackbody radiation, does not show Bose-Einstein condensation. Instead of massively populating the cavity ground mode, photons vanish in the cavity walls when they are cooled down. The situation is different in an ultrashort optical cavity im- printing a low-frequency cutoff on the photon energy spectrum that is well above the thermal energy. The latter allows for a thermalization process in which both tempera- ture and photon number can be tuned independently of each other or, correspondingly, for a non-vanishing photon chemical potential. We here describe experiments demon- strating the fluorescence-induced thermalization and Bose-Einstein condensation of a two-dimensional photon gas in the dye microcavity. Moreover, recent measurements on the photon statistics of the condensate, showing Bose-Einstein condensation in the grandcanonical ensemble limit, will be reviewed. 1 Introduction Quantum statistical effects become relevant when a gas of particles is cooled, or its den- sity is increased, to the point where the associated de Broglie wavepackets spatially over- arXiv:1611.10286v1 [cond-mat.quant-gas] 30 Nov 2016 lap. For particles with integer spin (bosons), the phenomenon of Bose-Einstein condensation (BEC) then leads to macroscopic occupation of a single quantum state at finite tempera- tures [1]. -
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 Equationsofstate .......................... 5 1.2 Lawsofthermodynamics. 7 1.3 Maxwell relations and thermodynamic derivatives . .... 9 1.4 Specificheatsandcompressibilities . 10 1.5 Thermodynamicpotentials . 12 1.6 Fluctuations and thermodynamic stability . .. 15 1.7 Phasetransitions ........................... 16 1.8 EntropyandGibbsParadox. 18 2 Non-Interacting Particles 23 1 2.1 Spin- 2 particlesinamagneticfield . 23 2.2 Maxwell-Boltzmannstatistics . 28 2.3 Idealgas................................ 32 2.4 Fermi-Diracstatistics. 35 2.5 Bose-Einsteinstatistics. 36 3 Statistical Ensembles 39 3.1 Ensemblesinphasespace . 39 3.2 Liouville’stheorem . .. .. .. .. .. .. .. .. .. .. 42 3.3 Microcanonicalensembles . 45 3.4 Free particles and multi-dimensional spheres . .... 48 3.5 Canonicalensembles . 50 3.6 Grandcanonicalensembles . 54 3.7 Isobaricensembles .......................... 58 3.8 Informationtheory . .. .. .. .. .. .. .. .. .. .. 62 4 Real Gases and Liquids 67 4.1 Correlationfunctions. 67 4.2 Thevirialtheorem .......................... 73 4.3 Mean field theory for the van der Waals equation . 76 4.4 Osmosis ................................ 80 3 4 CONTENTS 5 Quantum Gases and Liquids 83 5.1 Statisticsofidenticalparticles. .. 83 5.2 Blackbodyradiationandthephotongas . 88 5.3 Phonons and the Debye theory of specific heats . 96 5.4 Bosonsatnon-zerochemicalpotential . -
Grand-Canonical Ensemble
PHYS4006: Thermal and Statistical Physics Lecture Notes (Unit - IV) Open System: Grand-canonical Ensemble Dr. Neelabh Srivastava (Assistant Professor) Department of Physics Programme: M.Sc. Physics Mahatma Gandhi Central University Semester: 2nd Motihari-845401, Bihar E-mail: [email protected] • In microcanonical ensemble, each system contains same fixed energy as well as same number of particles. Hence, the system dealt within this ensemble is a closed isolated system. • With microcanonical ensemble, we can not deal with the systems that are kept in contact with a heat reservoir at a given temperature. 2 • In canonical ensemble, the condition of constant energy is relaxed and the system is allowed to exchange energy but not the particles with the system, i.e. those systems which are not isolated but are in contact with a heat reservoir. • This model could not be applied to those processes in which number of particle varies, i.e. chemical process, nuclear reactions (where particles are created and destroyed) and quantum process. 3 • So, for the method of ensemble to be applicable to such processes where number of particles as well as energy of the system changes, it is necessary to relax the condition of fixed number of particles. 4 • Such an ensemble where both the energy as well as number of particles can be exchanged with the heat reservoir is called Grand Canonical Ensemble. • In canonical ensemble, T, V and N are independent variables. Whereas, in grand canonical ensemble, the system is described by its temperature (T),volume (V) and chemical potential (μ). 5 • Since, the system is not isolated, its microstates are not equally probable. -
8.1. Spin & Statistics
8.1. Spin & Statistics Ref: A.Khare,”Fractional Statistics & Quantum Theory”, 1997, Chap.2. Anyons = Particles obeying fractional statistics. Particle statistics is determined by the phase factor eiα picked up by the wave function under the interchange of the positions of any pair of (identical) particles in the system. Before the discovery of the anyons, this particle interchange (or exchange) was treated as the permutation of particle labels. Let P be the operator for this interchange. P2 =I → e2iα =1 ∴ eiα = ±1 i.e., α=0,π Thus, there’re only 2 kinds of statistics, α=0(π) for Bosons (Fermions) obeying Bose-Einstein (Fermi-Dirac) statistics. Pauli’s spin-statistics theorem then relates particle spin with statistics, namely, bosons (fermions) are particles with integer (half-integer) spin. To account for the anyons, particle exchange is re-defined as an observable adiabatic (constant energy) process of physically interchanging particles. ( This is in line with the quantum philosophy that only observables are physically relevant. ) As will be shown later, the new definition does not affect statistics in 3-D space. However, for particles in 2-D space, α can be any (real) value; hence anyons. The converse of the spin-statistic theorem then implies arbitrary spin for 2-D particles. Quantization of S in 3-D See M.Alonso, H.Valk, “Quantum Mechanics: Principles & Applications”, 1973, §6.2. In 3-D, the (spin) angular momentum S has 3 non-commuting components satisfying S i,S j =iℏε i j k Sk 2 & S ,S i=0 2 This means a state can be the simultaneous eigenstate of S & at most one Si. -
The Grand Canonical Ensemble
University of Central Arkansas The Grand Canonical Ensemble Stephen R. Addison Directory ² Table of Contents ² Begin Article Copyright °c 2001 [email protected] Last Revision Date: April 10, 2001 Version 0.1 Table of Contents 1. Systems with Variable Particle Numbers 2. Review of the Ensembles 2.1. Microcanonical Ensemble 2.2. Canonical Ensemble 2.3. Grand Canonical Ensemble 3. Average Values on the Grand Canonical Ensemble 3.1. Average Number of Particles in a System 4. The Grand Canonical Ensemble and Thermodynamics 5. Legendre Transforms 5.1. Legendre Transforms for two variables 5.2. Helmholtz Free Energy as a Legendre Transform 6. Legendre Transforms and the Grand Canonical Ensem- ble 7. Solving Problems on the Grand Canonical Ensemble Section 1: Systems with Variable Particle Numbers 3 1. Systems with Variable Particle Numbers We have developed an expression for the partition function of an ideal gas. Toc JJ II J I Back J Doc Doc I Section 2: Review of the Ensembles 4 2. Review of the Ensembles 2.1. Microcanonical Ensemble The system is isolated. This is the ¯rst bridge or route between mechanics and thermodynamics, it is called the adiabatic bridge. E; V; N are ¯xed S = k ln (E; V; N) Toc JJ II J I Back J Doc Doc I Section 2: Review of the Ensembles 5 2.2. Canonical Ensemble System in contact with a heat bath. This is the second bridge between mechanics and thermodynamics, it is called the isothermal bridge. This bridge is more elegant and more easily crossed. T; V; N ¯xed, E fluctuates. -
Many Particle Orbits – Statistics and Second Quantization
H. Kleinert, PATH INTEGRALS March 24, 2013 (/home/kleinert/kleinert/books/pathis/pthic7.tex) Mirum, quod divina natura dedit agros It’s wonderful that divine nature has given us fields Varro (116 BC–27BC) 7 Many Particle Orbits – Statistics and Second Quantization Realistic physical systems usually contain groups of identical particles such as spe- cific atoms or electrons. Focusing on a single group, we shall label their orbits by x(ν)(t) with ν =1, 2, 3, . , N. Their Hamiltonian is invariant under the group of all N! permutations of the orbital indices ν. Their Schr¨odinger wave functions can then be classified according to the irreducible representations of the permutation group. Not all possible representations occur in nature. In more than two space dimen- sions, there exists a superselection rule, whose origin is yet to be explained, which eliminates all complicated representations and allows only for the two simplest ones to be realized: those with complete symmetry and those with complete antisymme- try. Particles which appear always with symmetric wave functions are called bosons. They all carry an integer-valued spin. Particles with antisymmetric wave functions are called fermions1 and carry a spin whose value is half-integer. The symmetric and antisymmetric wave functions give rise to the characteristic statistical behavior of fermions and bosons. Electrons, for example, being spin-1/2 particles, appear only in antisymmetric wave functions. The antisymmetry is the origin of the famous Pauli exclusion principle, allowing only a single particle of a definite spin orientation in a quantum state, which is the principal reason for the existence of the periodic system of elements, and thus of matter in general. -
The Conventionality of Parastatistics
The Conventionality of Parastatistics David John Baker Hans Halvorson Noel Swanson∗ March 6, 2014 Abstract Nature seems to be such that we can describe it accurately with quantum theories of bosons and fermions alone, without resort to parastatistics. This has been seen as a deep mystery: paraparticles make perfect physical sense, so why don't we see them in nature? We consider one potential answer: every paraparticle theory is physically equivalent to some theory of bosons or fermions, making the absence of paraparticles in our theories a matter of convention rather than a mysterious empirical discovery. We argue that this equivalence thesis holds in all physically admissible quantum field theories falling under the domain of the rigorous Doplicher-Haag-Roberts approach to superselection rules. Inadmissible parastatistical theories are ruled out by a locality- inspired principle we call Charge Recombination. Contents 1 Introduction 2 2 Paraparticles in Quantum Theory 6 ∗This work is fully collaborative. Authors are listed in alphabetical order. 1 3 Theoretical Equivalence 11 3.1 Field systems in AQFT . 13 3.2 Equivalence of field systems . 17 4 A Brief History of the Equivalence Thesis 20 4.1 The Green Decomposition . 20 4.2 Klein Transformations . 21 4.3 The Argument of Dr¨uhl,Haag, and Roberts . 24 4.4 The Doplicher-Roberts Reconstruction Theorem . 26 5 Sharpening the Thesis 29 6 Discussion 36 6.1 Interpretations of QM . 44 6.2 Structuralism and Haecceities . 46 6.3 Paraquark Theories . 48 1 Introduction Our most fundamental theories of matter provide a highly accurate description of subatomic particles and their behavior. -
Equipartition of Energy
Equipartition of Energy The number of degrees of freedom can be defined as the minimum number of independent coordinates, which can specify the configuration of the system completely. (A degree of freedom of a system is a formal description of a parameter that contributes to the state of a physical system.) The position of a rigid body in space is defined by three components of translation and three components of rotation, which means that it has six degrees of freedom. The degree of freedom of a system can be viewed as the minimum number of coordinates required to specify a configuration. Applying this definition, we have: • For a single particle in a plane two coordinates define its location so it has two degrees of freedom; • A single particle in space requires three coordinates so it has three degrees of freedom; • Two particles in space have a combined six degrees of freedom; • If two particles in space are constrained to maintain a constant distance from each other, such as in the case of a diatomic molecule, then the six coordinates must satisfy a single constraint equation defined by the distance formula. This reduces the degree of freedom of the system to five, because the distance formula can be used to solve for the remaining coordinate once the other five are specified. The equipartition theorem relates the temperature of a system with its average energies. The original idea of equipartition was that, in thermal equilibrium, energy is shared equally among all of its various forms; for example, the average kinetic energy per degree of freedom in the translational motion of a molecule should equal that of its rotational motions. -
Microcanonical, Canonical, and Grand Canonical Ensembles Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: September 30, 2016)
The equivalence: microcanonical, canonical, and grand canonical ensembles Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: September 30, 2016) Here we show the equivalence of three ensembles; micro canonical ensemble, canonical ensemble, and grand canonical ensemble. The neglect for the condition of constant energy in canonical ensemble and the neglect of the condition for constant energy and constant particle number can be possible by introducing the density of states multiplied by the weight factors [Boltzmann factor (canonical ensemble) and the Gibbs factor (grand canonical ensemble)]. The introduction of such factors make it much easier for one to calculate the thermodynamic properties. ((Microcanonical ensemble)) In the micro canonical ensemble, the macroscopic system can be specified by using variables N, E, and V. These are convenient variables which are closely related to the classical mechanics. The density of states (N E,, V ) plays a significant role in deriving the thermodynamic properties such as entropy and internal energy. It depends on N, E, and V. Note that there are two constraints. The macroscopic quantity N (the number of particles) should be kept constant. The total energy E should be also kept constant. Because of these constraints, in general it is difficult to evaluate the density of states. ((Canonical ensemble)) In order to avoid such a difficulty, the concept of the canonical ensemble is introduced. The calculation become simpler than that for the micro canonical ensemble since the condition for the constant energy is neglected. In the canonical ensemble, the system is specified by three variables ( N, T, V), instead of N, E, V in the micro canonical ensemble. -
Fastest Frozen Temperature for a Thermodynamic System
Fastest Frozen Temperature for a Thermodynamic System X. Y. Zhou, Z. Q. Yang, X. R. Tang, X. Wang1 and Q. H. Liu1, 2, ∗ 1School for Theoretical Physics, School of Physics and Electronics, Hunan University, Changsha, 410082, China 2Synergetic Innovation Center for Quantum Effects and Applications (SICQEA), Hunan Normal University,Changsha 410081, China For a thermodynamic system obeying both the equipartition theorem in high temperature and the third law in low temperature, the curve showing relationship between the specific heat and the temperature has two common behaviors: it terminates at zero when the temperature is zero Kelvin and converges to a constant value of specific heat as temperature is higher and higher. Since it is always possible to find the characteristic temperature TC to mark the excited temperature as the specific heat almost reaches the equipartition value, it is also reasonable to find a temperature in low temperature interval, complementary to TC . The present study reports a possibly universal existence of the such a temperature #, defined by that at which the specific heat falls fastest along with decrease of the temperature. For the Debye model of solids, below the temperature # the Debye's law manifest itself. PACS numbers: I. INTRODUCTION A thermodynamic system usually obeys two laws in opposite limits of temperature; in high temperature the equipar- tition theorem works and in low temperature the third law of thermodynamics holds. From the shape of a curve showing relationship between the specific heat and the temperature, the behaviors in these two limits are qualitatively clear: in high temperature it approaches to a constant whereas it goes over to zero during the temperature is lowering to the zero Kelvin. -
Exercise 18.4 the Ideal Gas Law Derived
9/13/2015 Ch 18 HW Ch 18 HW Due: 11:59pm on Monday, September 14, 2015 To understand how points are awarded, read the Grading Policy for this assignment. Exercise 18.4 ∘ A 3.00L tank contains air at 3.00 atm and 20.0 C. The tank is sealed and cooled until the pressure is 1.00 atm. Part A What is the temperature then in degrees Celsius? Assume that the volume of the tank is constant. ANSWER: ∘ T = 175 C Correct Part B If the temperature is kept at the value found in part A and the gas is compressed, what is the volume when the pressure again becomes 3.00 atm? ANSWER: V = 1.00 L Correct The Ideal Gas Law Derived The ideal gas law, discovered experimentally, is an equation of state that relates the observable state variables of the gaspressure, temperature, and density (or quantity per volume): pV = NkBT (or pV = nRT ), where N is the number of atoms, n is the number of moles, and R and kB are ideal gas constants such that R = NA kB, where NA is Avogadro's number. In this problem, you should use Boltzmann's constant instead of the gas constant R. Remarkably, the pressure does not depend on the mass of the gas particles. Why don't heavier gas particles generate more pressure? This puzzle was explained by making a key assumption about the connection between the microscopic world and the macroscopic temperature T . This assumption is called the Equipartition Theorem. The Equipartition Theorem states that the average energy associated with each degree of freedom in a system at T 1 k T k = 1.38 × 10−23 J/K absolute temperature is 2 B , where B is Boltzmann's constant.