DOCSLIB.ORG

4. Classical Phase Space 4.1. Phase Space and Probability Density

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
4. Classical Phase Space 4.1. Phase Space and Probability Density 4. Classical phase space quantum states in the part of the space under consideration. 4.1. Phase space and probability density The ensemble or statistical set consists, at a given We consider a system of N particles in a d-dimensional moment, of all those phase space points which correspond space. Canonical coordinates and momenta to a given macroscopic system. Corresponding to a macro(scopic) state of the system q = (q1,...,qdN ) there are thus a set of micro(scopic) states which belong p = (p1,...,pdN ) to the ensemble with the probability ρ(P ) dΓ. ρ(P ) is the probability density which is normalized to unity: determine exactly the microscopic state of the system. The phase space is the 2dN-dimensional space (p, q) , dΓ ρ(P )=1, whose every point P = (p, q) corresponds to a possible{ } Z state of the system. A trajectory is such a curve in the phase space along and it gives the local density of points in (q,p)-space at which the point P (t) as a function of time moves. time t. Trajectories are determined by the classical equations of The statistical average, or the ensemble expectation motion value, of a measurable quantity f = f(P ) is dq ∂H i = f = dΓ f(P )ρ(P ). dt ∂p h i i Z dp ∂H i = , We associate every phase space point with the velocity dt ∂q − i field ∂H ∂H where V = (q, ˙ p˙)= , . ∂p − ∂q H = H(q ,...,q ,p ,...,p ,t) 1 dN 1 dN The probability current is then Vρ. The probability = H(q,p,t)= H(P,t) weight of an element Γ0 evolves then like is the Hamiltonian function of the system. ∂ The trajectory is stationary, if H does not depend on ρ dΓ= Vρ dS. ∂t Γ − ∂Γ · time: trajectories starting from the same initial point P Z 0 Z 0 are identical. d S Trajectories cannot cross: if trajectories meet at a point, they must be identical! Let F = F (q,p,t) be a property of the system. Now G 0 dF ∂F = + F, H , dt ∂t { } Because where F, G stands for Poisson brackets { } Vρ dS = (Vρ) dΓ, ∂Γ · Γ ∇ · ∂F ∂G ∂G ∂F Z 0 Z 0 F, G . continuity equation { }≡ ∂q ∂p − ∂q ∂p we get in the limit Γ0 0 the i i i i i → X ∂ We define the volume measure of the phase space ρ + (Vρ)=0. ∂t ∇ · dN dqidpi dN According to the equations of motion dΓ= = h− dq dq dp dp . h 1 · · · dN 1 · · · dN i=1 Y ∂H q˙i = 34 Here h =6.62608 10− Js is the Planck constant. (Often ∂pi dΓ has 1/N! in front· to remove degeneracy caused by ∂H p˙i = identical particles.) − ∂qi Note: [dq dp]= Js, so dΓ is dimensionless. Note: In classical dynamics the normalization is we have ∂q˙ ∂p˙ irrelevant. In quantum mechanics, it is natural to choose i + i =0, ∂q ∂p h = Planck constant. i i Note: ∆0Γ = 1 corresponds to the smallest possible so we end up with the incompressibility condition volume element of the phase space where a point (sourceless) representing the system can be localized in accordance ∂q˙ ∂p˙ with the QM uncertainty principle. The volume V = i + i =0. ∇ · ∂q ∂p ∆Γ = dΓ is then roughly equal to the number of i i i X R 1 From the continuity equation we get then Ergodic flow: Almost all points of the surface ΓE are sometimes arbitrarily close to any point in ∆ΓE . ∂ρ 0 = + (Vρ) ∂t ∇ · ⇔ ∂ρ The flow is ergodic if f(P ), f(P ) ”smooth enough”, = + ρ V + V ρ ∀ ∂t ∇ · · ∇ ¯ ∂ρ f = f E = + V ρ. h i ∂t · ∇ holds. Here f¯ is the time average When we employ the convective time derivative 1 T f¯ = lim dt f(P (t)) d ∂ T T = + V →∞ Z0 dt ∂t · ∇ and f the energy surface expectation value ∂ ∂ ∂ E = + q˙ +p ˙ , h i ∂t i ∂q i ∂p i i i 1 X f E = dΓE f(P ). h i ΣE the continuity equation can be written in the form known Z as the Liouville theorem We define the microcanonical ensemble so that its density distribution is d ρ(P (t),t)=0. dt 1 ρE(P )= δ(H(P ) E). ΣE − Thus, the points in the phase space move like an incompressible fluid, and the local probability density ρ Every point of the energy surface belongs with the same remains constant during evolution. Different points probability to the microcanonical ensemble. ∂ρE naturally can have different ρ. The microcanonical ensemble is stationary, i.e. ∂t =0 and the expectation values over it temporal constants. 4.2. Flow in phase space The mixing flow is such an ergodic flow where the points of an energy surface element dΓ disperse in the The (constant) energy surface ΓE is the manifold E determined by the equation course of time all over the energy surface. Ifρ ˆE(P,t) is an arbitrary non stationary density distribution at the H(q,p)= E. moment t = t0, then 1 If the energy is a constant of motion, every phase point lim ρˆE(P,t)= δ(H(P ) E)= ρE(P ) i t ΣE − P (t) moves on a certain energy surface ΓEi, of dim. →∞ (2Nd 1). and The expectation− value of the energy of the system lim f = lim dΓˆρE(P,t)f(P ) t h i t E = H = dΓ Hρ →∞ →∞ Z h i Z = dΓ f(P )ρE(P ) is also a constant of motion. Z = f The volume of the energy surface is h iE i.e. the density describing an arbitrary (non equilibrium) Σ = dΓ = dΓ δ(H(P ) E). state evolves towards a microcanonical ensemble under E E − Z Z mixing flow. The volume of the phase space is Liouville theorem (volume element(s) are conserved) + mixing: a phase space infinitesimal volume element ∞ becomes infinitely stretched and folded and is distributed dΓ= dE ΣE. over the full energy surface. Z Z−∞ Let us consider the time evolution of a surface element 4.3. Microcanonical ensemble and ∆ΓE of an energy surface. entropy If the total energy of a macroscopic system is known Non-ergodic flow: In the course of time the element exactly and it remains fixed, the equilibrium state can be ∆ΓE traverses only a part of the whole energy described by a microcanonical ensemble. The surface ΓE. corresponding probability density is Examples: periodic motion; presence of other 1 constants of motion. ρE(P )= δ(H(P ) E). ΣE − 2 For a convenience we allow the energy to have some A) Let us now show that the maximisation of the Gibbs ”tolerance” and define entropy gives us microcanonical ensemble, when the energy of the system is restricted to (E, E + ∆E): 1 ρE,∆E(P )= θ(E + ∆E H(P ))θ(H(P ) E). ZE,∆E − − δS = k dΓ(δρ ln ρ + ρδ ln ρ) − B Here the normalization constant Z∆ΓE = k dΓδρ(ln ρ +1)=0. − B Z = dΓ θ(E + ∆E H(P ))θ(H(P ) E) ∆ΓE E,∆E − − Z Z Because δ1= dΓδρ = 0, above condition is satisfied if is the microcanonical state sum or partition function. ln ρ =const. or ρ(P ) =const., when P ∆Γ . R ∈ E ZE,∆E is the number of states contained in the energy S is indeed maximum: slice E<H<E + ∆E (see the volume measure of the phase space). In the microcanonical ensemble the 1 ∂2(ρ ln ρ) δ2S = k dΓ (δρ)2 probability is distributed evenly in every allowed part of − B 2 ∂ρ2 Z∆ΓE the phase space. 1 2 1 = kB dΓ (δρ) 0. − ∆Γ 2 ρ ≤ Entropy Z E We define statistical entropy as the quantity (depending B) Additive: if we consider 2 separate systems, the joint on ρ) which is a) maximized for a physical ensemble and probability distribution is ρ12(P1, P2)= ρ1(P1)ρ2(P2), b) is an extensive quantity, in order to connect to the laws and dΓ12 = dΓ1dΓ2. of thermodynamics. The maximization of the entropy determines the physical distribution ρ. The Gibbs entropy S = k dΓ ρ ln ρ 12 − B 12 12 12 Z S = k dΓ ρ(P ) ln ρ(P ) = kB dΓ1dΓ2 ρ1ρ2(ln ρ1 + ln ρ2) − B − Z Z = k ( dΓ ρ ln ρ + dΓ ρ ln ρ )= S + S has these properties, as we shall show shortly. − B 1 1 1 2 2 2 1 2 Z Z Let ∆Γi the volume of the phase space element i and ρi the average probability density in i. The state of the system is, with the probability Entropy and disorder Maximizing the entropy S minimization of the ⇒ pi = ρi ∆Γi, information about the system. In a microcanonical ensemble complete lack of information (besides the total in the element i and energy) means that all states with the same total energy are equally probable. pi =1. Conversely, losing the information (= non-trivial X probability distribution) maximizing the entropy. We choose the sizes of all elements to be smallest Maximum of entropy maximum⇒ of disorder. ⇔ possible, i.e. ∆Γi = 1. Then S = k ∆Γ ρ ln ρ = k ρ ∆Γ ln ρ ∆Γ − B i i i − B i i i i i i X X = k p ln p , − B i i i X since ln ∆Γi =0. If ρ is smooth in the range ∆Γ = W we have 1 ρ = , W so that 1 1 S = k ln dΓ. − B W W Z We end up with the Boltzmann entropy S = kB ln W.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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].
    [Show full text]
  • 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 .
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Statistical Physics 06/07
    STATISTICAL PHYSICS 06/07 Quantum Statistical Mechanics Tutorial Sheet 3 The questions that follow on this and succeeding sheets are an integral part of this course. The code beside each question has the following significance: • K: key question – explores core material • R: review question – an invitation to consolidate • C: challenge question – going beyond the basic framework of the course • S: standard question – general fitness training! 3.1 Particle Number Fluctuations for Fermions [s] (a) For a single fermion state in the grand canonical ensemble, show that 2 h(∆nj) i =n ¯j(1 − n¯j) wheren ¯j is the mean occupancy. Hint: You only need to use the exclusion principle not the explict form ofn ¯j. 2 How is the fact that h(∆nj) i is not in general small compared ton ¯j to be reconciled with the sharp values of macroscopic observables? (b) For a gas of noninteracting particles in the grand canonical ensemble, show that 2 X 2 h(∆N) i = h(∆nj) i j (you will need to invoke that nj and nk are uncorrelated in the GCE for j 6= k). Hence show that for noninteracting Fermions Z h(∆N)2i = g() f(1 − f) d follows from (a) where f(, µ) is the F-D distribution and g() is the density of states. (c) Show that for low temperatures f(1 − f) is sharply peaked at = µ, and hence that 2 h∆N i ' kBT g(F ) where F = µ(T = 0) R ∞ ex dx [You may use without proof the result that −∞ (ex+1)2 = 1.] 3.2 Entropy of the Ideal Fermi Gas [C] The Grand Potential for an ideal Fermi is given by X Φ = −kT ln [1 + exp β(µ − j)] j Show that for Fermions X Φ = kT ln(1 − pj) , j where pj = f(j) is the probability of occupation of the state j.
    [Show full text]
  • 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 degrees of freedom are described quantum mechanically and particles can interchange their locations – in such cases we can write the expression for the canonical partition function, but because of the restriction on the occupation numbers we simply cannot calculate it! (see end of previous write-up). Even for classical systems, we do not know how to deal with problems where the number of particles is not fixed (open systems). For example, suppose we have a surface on which certain types of atoms can be adsorbed (trapped). The surface is in contact with a gas containing these atoms, and depending on conditions some will stick to the surface while other become free and go into the gas. Suppose we are interested only in the properties of the surface (for instance, the average number of trapped atoms as a function of temperature). Since the numbers of atoms on the surface varies, this is an open system and we still do not know how to solve this problem. So for these reasons we need to introduce grand-canonical ensembles. This will finally allow us to study quantum ideal gases (our main goal for this course). As we expect, the results we’ll obtain at high temperatures will agree with the classical predictions we already have, however, as we will see, the low-temperature quantum behavior is really interesting and worth the effort! Like we did for canonical ensembles, I’ll introduce the formalism for classical problems first, and then we’ll generalize to quantum systems.
    [Show full text]
  • 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.
    [Show full text]
  • Quantum Statistics (Quantenstatistik)
    Chapter 7 Quantum Statistics (Quantenstatistik) 7.1 Density of states of massive particles Our goal: discover what quantum effects bring into the statistical description of systems. Formulate conditions under which QM-specific effects are not essential and classical descriptions can be used. Consider a single particle in a box with infinitely high walls U ε3 ε2 Figure 7.1: Levels of a single particle in a 3D box. At the ε1 box walls the potential is infinite and the wave-function is L zero. A one-particle Hamiltonian has only the momentum p^2 ~2 H^ = = − r2 (7.1) 2m 2m The infinite potential at the box boundaries imposes the zero-wave-function boundary condition. Hence the eigenfunctions of the Hamiltonian are k = sin(kxx) sin(kyy) sin(kzz) (7.2) ~ 2π + k = ~n; ~n = (n1; n2; n3); ni 2 Z (7.3) L ( ) ~ ~ 2π 2 ~p = ~~k; ϵ = k2 = ~n2;V = L3 (7.4) 2m 2m L 4π2~2 ) ϵ = ~n2 (7.5) 2mV 2=3 Let us introduce spin s (it takes into account possible degeneracy of the energy levels) and change the summation over ~n to an integration over ϵ X X X X X X Z X Z 1 ··· = ··· = · · · ∼ d3~n ··· = dn4πn2 ::: (7.6) 0 ν s ~k s ~n s s 45 46 CHAPTER 7. QUANTUM STATISTICS (QUANTENSTATISTIK) (2mϵ)1=2 V 1=3 1 1 V m3=2 n = V 1=3 ; dn = 2mdϵ, 4πn2dn = p ϵ1=2dϵ (7.7) 2π~ 2π~ 2 (2mϵ)1=2 2π2~3 X = gs = 2s + 1 spin-degeneracy (7.8) s Z X g V m3=2 1 ··· = ps dϵϵ1=2 ::: (7.9) 2~3 ν 2π 0 Number of states in a given energy range is higher at high energies than at low energies.
    [Show full text]
  • LECTURE 9 Statistical Mechanics Basic Methods We Have Talked
    LECTURE 9 Statistical Mechanics Basic Methods We have talked about ensembles being large collections of copies or clones of a system with some features being identical among all the copies. There are three different types of ensembles in statistical mechanics. 1. If the system under consideration is isolated, i.e., not interacting with any other system, then the ensemble is called the microcanonical ensemble. In this case the energy of the system is a constant. 2. If the system under consideration is in thermal equilibrium with a heat reservoir at temperature T , then the ensemble is called a canonical ensemble. In this case the energy of the system is not a constant; the temperature is constant. 3. If the system under consideration is in contact with both a heat reservoir and a particle reservoir, then the ensemble is called a grand canonical ensemble. In this case the energy and particle number of the system are not constant; the temperature and the chemical potential are constant. The chemical potential is the energy required to add a particle to the system. The most common ensemble encountered in doing statistical mechanics is the canonical ensemble. We will explore many examples of the canonical ensemble. The grand canon- ical ensemble is used in dealing with quantum systems. The microcanonical ensemble is not used much because of the difficulty in identifying and evaluating the accessible microstates, but we will explore one simple system (the ideal gas) as an example of the microcanonical ensemble. Microcanonical Ensemble Consider an isolated system described by an energy in the range between E and E + δE, and similar appropriate ranges for external parameters xα.
    [Show full text]
  • Statistical Physics Syllabus Lectures and Recitations
    Graduate Statistical Physics Syllabus Lectures and Recitations Lectures will be held on Tuesdays and Thursdays from 9:30 am until 11:00 am via Zoom: https://nyu.zoom.us/j/99702917703. Recitations will be held on Fridays from 3:30 pm until 4:45 pm via Zoom. Recitations will begin the second week of the course. David Grier's office hours will be held on Mondays from 1:30 pm to 3:00 pm in Physics 873. Ankit Vyas' office hours will be held on Fridays from 1:00 pm to 2:00 pm in Physics 940. Instructors David G. Grier Office: 726 Broadway, room 873 Phone: (212) 998-3713 email: [email protected] Ankit Vyas Office: 726 Broadway, room 965B Email: [email protected] Text Most graduate texts in statistical physics cover the material of this course. Suitable choices include: • Mehran Kardar, Statistical Physics of Particles (Cambridge University Press, 2007) ISBN 978-0-521-87342-0 (hardcover). • Mehran Kardar, Statistical Physics of Fields (Cambridge University Press, 2007) ISBN 978- 0-521-87341-3 (hardcover). • R. K. Pathria and Paul D. Beale, Statistical Mechanics (Elsevier, 2011) ISBN 978-0-12- 382188-1 (paperback). Undergraduate texts also may provide useful background material. Typical choices include • Frederick Reif, Fundamentals of Statistical and Thermal Physics (Waveland Press, 2009) ISBN 978-1-57766-612-7. • Charles Kittel and Herbert Kroemer, Thermal Physics (W. H. Freeman, 1980) ISBN 978- 0716710882. • Daniel V. Schroeder, An Introduction to Thermal Physics (Pearson, 1999) ISBN 978- 0201380279. Outline 1. Thermodynamics 2. Probability 3. Kinetic theory of gases 4.
    [Show full text]
  • Lecture 7: Ensembles
    Matthew Schwartz Statistical Mechanics, Spring 2019 Lecture 7: Ensembles 1 Introduction In statistical mechanics, we study the possible microstates of a system. We never know exactly which microstate the system is in. Nor do we care. We are interested only in the behavior of a system based on the possible microstates it could be, that share some macroscopic proporty (like volume V ; energy E, or number of particles N). The possible microstates a system could be in are known as the ensemble of states for a system. There are dierent kinds of ensembles. So far, we have been counting microstates with a xed number of particles N and a xed total energy E. We dened as the total number microstates for a system. That is (E; V ; N) = 1 (1) microstatesk withsaXmeN ;V ;E Then S = kBln is the entropy, and all other thermodynamic quantities follow from S. For an isolated system with N xed and E xed the ensemble is known as the microcanonical 1 @S ensemble. In the microcanonical ensemble, the temperature is a derived quantity, with T = @E . So far, we have only been using the microcanonical ensemble. 1 3 N For example, a gas of identical monatomic particles has (E; V ; N) V NE 2 . From this N! we computed the entropy S = kBln which at large N reduces to the Sackur-Tetrode formula. 1 @S 3 NkB 3 The temperature is T = @E = 2 E so that E = 2 NkBT . Also in the microcanonical ensemble we observed that the number of states for which the energy of one degree of freedom is xed to "i is (E "i) "i/k T (E " ).
    [Show full text]