5. Systems in contact with a thermal bath

Size: px
Start display at page:

Download "5. Systems in contact with a thermal bath"

Transcription

1 5. Systems in contact with a thermal bath So far, isolated systems (micro-canonical methods) 5.1 Constant number of particles:kittel&kroemer Chap. 3 Boltzmann factor Partition function (canonical methods) Ideal gas (again) Entropy Sackur-Tetrode formula Chemical potential 5.2 Exchange of particles: Kittel&Kroemer Chap. 5 Gibbs Factor Grand Partition Function or Gibbs sum 5.3 Fluctuations (grand canonical methods) Phys 112 (S2014) 05 Boltzmann-Gibbs 1

2 Ideal Gas Thermal Bath Consider 1 particle in thermal contact with a thermal bath made of an ideal gas with a large number M of particles We assume that the overall system is isolated and in equilibrium at temperature. We call U the total energy (which is constant). We would like to compute the probability of having our particle in a state of energy. Because in equilibrium, all states of the overall system are equiprobable, this is equivalent to computing the number of states such that our particle has energy and the thermal bath EU-. Using the fact that for an ideal gas of energy E and particle number N p, the g number of states is Res exp g E 3/2 N p and that U 3 2 N p we obtain for large N p prob ( U ) 3 2 N p N p 3 2 N p N p exp Phys 112 (S2014) 05 Boltzmann-Gibbs 2

3 R, U o - # States in configuration 1 g R U o Fundamental postulate (or H theorem): at equilibrium Prob configuration Notes: exp s Boltzmann factor Zpartition function Boltzmann factor S, Probability of each state not identical < interaction with bath Total system Our system S + reservoir R isolated Configuration: 1 state of S + energy of reservoir g R ( U o ) g R ( U o 1) g R ( U o 2 ) exp σ R ( U o 1) σ R ( U o 2 ) exp σ R ( 2 1 ) U o exp 1 ( 2 1) Prob state 1 Prob state 2 Prob state 1 Prob state 2 Prob state s 1 Z exp s with Z all states s 1 exp Z exp s k s B T Phys 112 (S2014) 05 Boltzmann-Gibbs 3

4 Examples 2-state system Z 1+ exp exp U 1+ exp C V dq dt V (,N ) State 1 has energy 0 State 2 has energy U T exp + 1 V (,N ) U k b V (,N ) 1 Pr ob( 1) 1 + exp 0 2 exp k b k b T exp k b T k b T ( ) exp 1 Pr ob( 2) ( ) 1 + exp 1 + exp Prob(0) 1 C V Prob() T Phys 112 (S2014) 05 Boltzmann-Gibbs 4

5 Phys 112 (S2014) 05 Boltzmann-Gibbs Examples Free particle at temperature Prob Assumption: No additional degrees of freedom [ ] " $ # % [ ] " # energyy interval dv, p, p + dp $, dω volume element Momentum interval e.g. no spin, not a multi-atom molecule solid angle element Prob dv,, + d $ %$, dω volume element solid angle element Prob dv,[ v,v + dv] " $ # %$, dω volume element velocity interval solid angle element 1 Z exp p2 2M " $$ # $$ % Boltzmann factor does not vary appreciably over dp Maxwell-Boltzmann distribution KK p 392 2M 3/2 Z M 3 Z exp exp dv 1 2 Mv2 Normalizing to N particles, the probability to find one particle per unit volume in the velocity interval dv and solid angle dω is 3 M 2 Mv 2 f (v)dvdω n 2π exp 2 v2 dvdω with n N V 5 dvp 2 dpdω h 3 " $ # $ % number of spatial quantum states in phase space element ddω h 3 dvv 2 dvdω h 3 dω d cosθdφ

6 Another interpretation of the Boltzmann Distribution We derived it from number of states in the Reservoir But each excitation in the reservoir will have a typical energy Thermal fluctuations of the order of Difficult to get >> Prob exp Phys 112 (S2014) 05 Boltzmann-Gibbs 6

7 Definition all states s If states are degenerate in energy, each one has to be counted separately Relation with thermodynamic functions Energy U 1 Z Entropy Free Energy Chemical potential s Partition Function Z s exp s σ s p s F U σ log Z exp s logz log p s 1 s e s µ F log Z N,V N 2 logz s log e Z Z log Z + U ( log Z) σ F U 2 log Z 2 F as implied by the thermodynamic identity Phys 112 (S2014) 05 Boltzmann-Gibbs 7 T

8 Examples Energy of a two state system Z 1+ exp Free particle at temperature Assumption: No additional degrees of freedom e.g. no spin, not a multi-atom molecule Method 1: density in phase space Z 1 Energy: d 3 q d 3 p h 3 p x 2 + p y 2 + p z 2 2M exp V h 3 dp x e p x 2M # "## $ Z 1 V 2πM h 2 2 M V 2π 2 3 U 2 log Z 2πM 2 dp y e 2π 2 Method 2: quantum state in a box: see Kittel-Krömer (chapter 3 p 73) Same result, of course 3 2 VnQ with nq p y 2 2M exp 1+ exp M dp z e 3 2 p z 2 2 M exp log Z Phys 112 (S2014) 05 Boltzmann-Gibbs 8

9 Partition Function Power comes from summation methods Direct e.g. geometric series r i lim r i lim i0 m i0 Few term approximation useful if f decreases fast Integral approximation f ( i) f (x)dx i0 0 Other example: Harmonic oscillator Consider an harmonic oscillator ω Phys 112 (S2014) 05 Boltzmann-Gibbs 9 m m The probability of mode of energy sω, s integer 0 is Prob( s) 1 Z sω exp Z exp sω 1 s0 1 exp ω 1 r m+1 1 r 1 1 r f ( i) f ( 0) + f ( 1) + f ( 2) +... i0 log( Z ) U sω 2 ω exp ω 1

10 Applies to classical gases in equilibrium with no potential Start from Boltzmann distribution Prob (1 particle to be in state of energy ) with Z 1 d3 q d 3 p h 3 exp V M 2π 2 Change of variable to velocities: Probability of finding particle in v, v+dv, dω Maxwell distribution Maxwell Distribution K&K p. 392 For N particles, the volume density between p and p + d p N V p Mv Particle density distribution in velocity space Note: 1) Could get the factor in front directly as normalization 2) Note that no h factor Mv k BT 3) One can easily compute Equipartition Phys 112 (S2014) 05 Boltzmann-Gibbs 10 1 exp n Q d 3 p h 3 1 exp 3 Z 1 2 VnQ Probability of finding this particle in volume dv and between p and p + d p 1 exp > Vn Q n exp n Q d 3 p 1 n h 3 2π M 3 f (v)dvdω n M 2 Mv 2 exp 2π v 2 dvdω 2 3/2 dvd 3 p h 3 "#$ Number of states exp d 3 p

11 Individual States of probability p s Density of States ## "## $ + d Dd It is convenient to replace expressions such as by an integral over the energy : p s f s p s D s 0 p s f s (f s any function of the state s) s f s Dd d is the number of states between and + d and is called the density of states. We replace p s by p s s < s < +d Phys 112 (S2014) 05 Boltzmann-Gibbs D d since for small enough d, p s does not vary Note that if you ask what is the probability pd for the system to be between energy between and + d pd p s i.e., p D p s d For example, for Boltzmann distribution and a non relativistic ideal gas (monoatomic,spinless) p s ( ) 1 Z exp and using p 2M d 3 p p 2 dpd cosθdϕ p 2 dpdω Dd V d 3 x dω p 2 dp angles h 3 V 4π dp V 4π 1 p 2 d ( h 3 d h 3 2 2M )3/2 d V 4π M 3/2 d

12 Partition Function for N systems Weak interaction limit 2 sub systems in weak interaction: <>States of subsystems are not disturbed by interaction (just their population) Ψ 1+2 ( n 1,n 2 ) Ψ 1 ( n 1 )Ψ 2 ( n 2 ) with n 1, n 2 still meaningfull <> > Partition functions multiply for distinguishable systems Distinguishable systems Suppose first that we can distinguish each of the individual components of the system: e.g. adsorption sites on a solid, spins in a lattice For 2 subsystems Z exp s 1 + s2 Z ( 1 + 2) Z ( 1)Z ( 2) s s 1 2 and for N systems Z N ( Z ( 1) ) N Phys 112 (S2014) 05 Boltzmann-Gibbs 12

13 Examples Free energy of a paramagnetic system In a magnetic field B, mb if m is the magnetic moment. For 1 spin Z 1 exp mb + exp mb 2 cosh mb For N spins ZZ 1 N since they are distinguishable U 2 log Z Z 2 N cosh N mb NmB tanh mb Magnetization M U VB mb nm tanh M M m B mb Phys 112 (S2014) 05 Boltzmann-Gibbs 13

14 Indistinguishable Systems Indistinguishable:in quantum mechanics, free particles are indistinguishable We should count permutations of states among particles only once s 1 e.g. for 2 particles Z ( 1)Z ( 2) e e overcounts s 1,s 2 s 2,s 1 for s 1 s 2 we have to divide by 2 s 2 More generally if we have N identical systems, the Nth power of Z 1 can be expanded as N n a 1 n 1..a s n s...a m m n i n 1...n s...n m Therefore if we have N indistinguishable systems (a a s +...a m ) N s 1 s 2 If we have a large number of possible states, most of the terms are of the form Na i a j...a " # $# k This is a low occupation number approximation due to Gibbs ( no significant probability to have 2 particles in the same state: we will see how to treat the case of degenerate quantum gases) N terms corresponding to N different states which are singly occupied. If the multiple occupation of states is negligible we are indeed overcounting by N 1 ( N Z ( 1 )) N Z N Phys 112 (S2014) 05 Boltzmann-Gibbs 14

15 N atoms Indistinguishable Ideal Gas Z N 1 N Z 1 ( 1 )N ( N n QV) N V log N N log N N log Z N N log n Q + N log Z N N log n Q n + N with n N N V concentration M n Q 2π 2 Interpretation: n Q as quantum concentration de Broglie wave length 3 2 λ h Mv h 3M 1 2 Mv λ 3 3M h M 2π nq Phys 112 (S2014) 05 Boltzmann-Gibbs 15

16 Energy Free energy Pressure Entropy: Sackur Tetrode Ideal gas laws Phys 112 (S2014) 05 Boltzmann-Gibbs 16 U 2 log Z σ equipartition of energy 1/2 per degree of freedom if extra degrees of freedom, additional terms Well verified by experiment Involves quantum mechanics Chemical potential 3 2 N 3 2 Nk BT F log Z N log n Q N n p F V N V pv N Nk BT S F k B N log n Q + 5 n 2 S µ F N T dq 0 T N log n V Q identical to our expression counting states N N N with n Q log n n Q 3 M 2 2π 2

17 Barometric Equation: Three different ways µconstant Thermodynamic equilibrium between slabs at various z At constant temperature µ total C log n( z) Single particle occupancy of levels at different altitude n( z) prob( K,z) exp + Φ( z) K n 0 exp mgz Hydrodynamic equilibrium ( p + dp)da pda mgn ( z)dadz Generalizes to case when the temperature is not constant n Q pda ( p + dp)da mgn z pv N p( z) n z Φ( z) + C n z n z dadz 0 1 n z 1 n z n 0 exp mgz n z mg p z mg Phys 112 (S2014) 05 Boltzmann-Gibbs 17

18 Internal Degrees of Freedom (Optional) In general, we need to add internal degrees of freedom In most cases, the kinetic and internal degrees of freedom are independent, for one particle e.g. for multi-atomic molecules Rotation 10-2 ev Rotation energy j j( j + 1) 0 with multiplicity 2 j+1 Z int ( 2 j+1) exp j( j + 1) 0 j ( 2 j+1) exp j( j + 1) 0 1/2 dj exp( x) x 0 o dx with x j( j + 1) 0 dx ( 2 j + 1) 0 dj, x Z int exp o 4 o Vibration ev similar to harmonic oscillator o Spin in a magnetic field etc.. for >> 0 for >> 0 U 2 log Z C rot k B For < < 0, keep the first terms of the discrete sum Z int 1 + 3exp 2 o +... log Z int 3exp 2 o U 2 log Z 6 0 exp 2 o C rot 12k o b Z int exp nω 1 n0 1 exp ω ω for >>ω C vib k B Z Z Z kin int kinetic C V internal 2 exp 2 o 0-1/ j Diatomic molecules cf. KK fig 3.9 k B Rotation Slightly better approximation than 0 Vibration k Phys 112 (S2014) 05 Boltzmann-Gibbs 18 log

19 Exchanging Particles with an Ideal gas Let us consider again a system with an energy level, which can now be occupied by a variable number N of particles g g Res 1 e 5/2 We put in contact with a thermal reservoir constituting on a large number N p -N of free particles. Energy and particles can be exchanged with the reservoir. We consider a single state with N particles and an energy N We can compute the number of states of the overall system as a function of Ν and. M 2 U N 2π 2 3(N p N) N p N V e 5/2 1 + N N p N Phys 112 (S2014) 05 Boltzmann-Gibbs 19 5/2 3/2 N p N N p N M 2U 2π 2 3N p N p V Direct Counting < or Seckur Tetrode 3/2 N p N 1 N U 3/2 N p N g Res N exp µ N p e n N p 5/2 Q n exp N log n Q N U 3 n exp exp N ( µ ) lim 1+ a N N exp a N 2 N p b N exp N logb N

20 R, U o - Phys 112 (S2014) 05 Boltzmann-Gibbs Gibbs factor System in contact with thermal bath (reservoir) S, Exchange of particles # States in configuration 1 g R ( U o s, N oi N is ) Prob configuration At equilibrium Notes: Pr ob( state 1, N i 1 ) Pr ob( state 2, N i 2 ) Prob state s with, N is g R ( U o 1, N oi N i1 ) exp g R U o 2, N oi N i 2 Absolute activity λ i exp µ i Z grand partition function 20 Total system Our system + reservoir Configuration: 1 state s of S of energy s and # of particles N is + energy of reservoir g R ( U o s, N oi N is ) σ R 1 ( s, N is ) µ i N is Z exp i with Z all states s,n is s, N is exp µ i N is i U σ R 1 2 i N i N i1 N i 2 1 s, N is Z exp k B T µ i N is i

21 Definition Grand Partition Function Note: (s,ν) depends on N i both through through increase of N, and the interactions In weak interaction limit (ideal gas) and only 1 species, s, N s Z N s s Note If the N is N i are constant, all states s and occupation N is energy of each level Prob N s particles in state s exp µ i N i Z exp i Z ( N i ) the probabilities probabilities are the same as with Z This formalism is useful only if the N is change. i µ i N is s, N is Z exp µ s s,n s N s 1 Z exp ( µ s ) N s Phys 112 (S2014) 05 Boltzmann-Gibbs 21

22 Examples few state systems Kittel-Kroemer Gas adsorption (heme in myoglobin, hemoglobin, walls) Assume that each potential adsorption site can adsorb at most one gas molecule. What is the fraction of sites which are occupied? State 1 0 molecule > energy 0 State 2 1 molecule > energy If ideal (applies equally well to gas and dissolved gas and any low concentration species) Langmuir adsorption isotherm Fraction occupied f f Z 1 + exp µ 1 exp( µ ) +1 exp µ 1 + exp µ p n Q exp + p same form as Fermi- Dirac µ log n p log pv N n N gas n Q n Q V P Phys 112 (S2014) 05 Boltzmann-Gibbs 22

23 Ionized impurities in a semiconductor cf Kittel and Kroemer p370 Donor tends to have 1 more electron (e.g. P in Si) State 1 ionized D + : no electron > energy 0 State 2neutral 1 extra electron spin up > energy d State 3 neutral 1 extra electron spin down > energy d 1 Fraction ionized f exp d µ 2 spin states when donor is neutral d energy of electron on donor site Acceptor tends to have 1 fewer electron (e.g. Al in Si) State 1 ionized A - : one electron completing the bond> energy a State 2 1 neutral: missing a bond electron spin up > energy 0 2 spin states when acceptor is neutral a energy of electron on acceptor site µ determined by electron concentration in semiconductor Fermi level Phys 112 (S2014) 05 Boltzmann-Gibbs 23 Z 1+ 2exp µ d K&K use ionization energy I- d State 3 1 neutral: missing a bond electron spin down > energy 0 exp a µ Fraction 1 ionized f exp µ a exp µ a

24 Grand Partition Function Relation to thermodynamic functions Mean number of particles Entropy σ N i N is p s,nis p s,nis Energy Free Energy Grand Potential Ω defined as s N is s N is U 2 log Z log( p s,nis ) log Z s N is Z exp N is µ i N is i + log Z + µ i N i 2 i + F U σ log Z + µ i N i G F + pv i Ω F µ i N i i i µ i N is s, N is i µ i log Z µ i log Z log Z µ i,v V,µ i Phys 112 (S2014) 05 Boltzmann-Gibbs 24

25 Microscopic exchanges Fluctuations In a system in equilibrium, exchanges still go on at the microscopic level They are just balanced Macroscopic quantities sum of microscopic quantities N x mean s s Z But with a finite system, relative fluctuations on such sum is of the order 1/ N e.g., fluctuation on total energy Variance not entropy e U s p s s σ 2 E ( E E ) 2 E 2 E 2 2 e s s Z e s s s s Z Computation with partition function; By substitution we can see that σ 2 E E 2 E 2 2 U 2 2 log Z Similarly when there is exchange of particles N log Z log Z and σ 2 N N 2 N 2 µ N µ µ µ,v Phys 112 (S2014) 05 Boltzmann-Gibbs 25 s 2 U E 2 log Z

26 Fermi Dirac Examples Fermions particles with half integer spin Pauli exclusion principle: each state contains at most one particle For a state of energy Z 1 + exp µ N log Z σ N 2 µ,v N µ exp µ 1+ exp µ exp µ exp µ exp µ + 1 Prob( 1 particle) N 1 N 2 N Less fluctuation than a Poisson distribution Phys 112 (S2014) 05 Boltzmann-Gibbs 26

27 Bose Einstein Examples Bosons particles with integer spin For a state of energy [ Z exp µ ]N 1 N 1 exp µ N log Z µ,v σ N 2 N µ exp µ 1 exp µ exp µ exp µ 1 1 exp µ 1 N 1 + N 2 N More fluctuation than a Poisson distribution Phys 112 (S2014) 05 Boltzmann-Gibbs 27

28 Minimization of the Free Energy Consider a system in interaction with reservoir imposing temperature Constant exchange of energy (+ in some case volume, particles) > Fluctuations, probability distribution Contrary to a system which is able to loose energy to vacuum, system does not evolve to state of minimum energy (constantly kicked) Contrary to an isolated system, does not evolves to a configuration of maximum entropy (The entropy of the combination of the reservoir and the system is maximized) It evolves to a configuration of minimum Free Energy Balance between tendency to loose energy and to maximize entropy. More rigorously F L G L We cannot define the temperature of the system out of equilibrium. So we introduce ( U S ) U S R σ S ( U S ) Landau Free Energy (constant volume) ( U S,V s ) U S R σ S ( U S,V S ) + p R V S Landau Free Enthalpy (constant pressure) They are minimized for the most probable configuration( equilibrium) Can be generalized to any variable (Will be useful for phase transitions) Phys 112 (S2014) 05 Boltzmann-Gibbs 28

29 Landau Free Energy Let us consider a system S exchanging energy with a large reservoir R. (We are no more considering a single state for system S). Constant volume σ tot ( U S ) σ R U U S F L σ R U σ R U + σ S ( U S ) σ R U U + σ U S S ( S ) 1 ( U S ) R U S + σ S σ R ( U ) 1 F L ( U S ) with R ( U S ) U S R σ S ( U S ) Landau Free Energy R, U-U s Most probable configuration maximizes σ tot i.e. minimizes F L Note that this implies F L F L ( U S ) 0 σ S ( U S ) 1 U s U S R e.g. for a perfect gas 3/2 MU S 3π" σ S ( U S ) 2 N N log + 5 (Sackur Tetrode with N S 2U s 2 V 3N ) S,U S ( U S ) U S U S 3 2 N R Phys 112 (S2014) 05 Boltzmann-Gibbs 29

30 Exchange of Particles If we exchange particles, what is minimized is Ω L ( U S, N S ) U S R σ S ( U S, N S ) µ R N S Landau Grand Potential in a way similar to the previous slide σ tot ( U S, N S ) σ R U U S, N N S σ R U σ R U + σ S ( U S, N S ) σ R U U S σ R N N S + σ S ( U S, N S ) 1 ( U S, N S ) 1 U S + µ R N S + σ S R R with σ R U Ω L U S, N S R Most probable configuration maximizes σ tot i.e. minimizes Ω L Note that this implies Ω L ( U S, N S ) 0 σ S ( U S, N S ) 1 Ω L ( U S, N S ) 0 σ S ( U S, N S ) µ R U s U S R N s N S R ( U S, N S ) Phys 112 (S2014) 05 Boltzmann-Gibbs 30

5. Systems in contact with a thermal bath

5. Systems in contact with a thermal bath 5. Systems in contact with a thermal bath So far, isolated systems (micro-canonical methods) 5.1 Constant number of particles:kittel&kroemer Chap. 3 Boltzmann factor Partition function (canonical methods)

More information

4. Systems in contact with a thermal bath

4. Systems in contact with a thermal bath 4. Systems in contact with a thermal bath So far, isolated systems microcanonical methods 4.1 Constant number of particles:kittelkroemer Chap. 3 Boltzmann factor Partition function canonical methods Ideal

More information

Phase Transitions. Phys112 (S2012) 8 Phase Transitions 1

Phase Transitions. Phys112 (S2012) 8 Phase Transitions 1 Phase Transitions cf. Kittel and Krömer chap 10 Landau Free Energy/Enthalpy Second order phase transition Ferromagnetism First order phase transition Van der Waals Clausius Clapeyron coexistence curve

More information

Chapter 5. Chemical potential and Gibbs distribution

Chapter 5. Chemical potential and Gibbs distribution Chapter 5 Chemical potential and Gibbs distribution 1 Chemical potential So far we have only considered systems in contact that are allowed to exchange heat, ie systems in thermal contact with one another

More information

Part II: Statistical Physics

Part II: Statistical Physics Chapter 7: Quantum Statistics SDSMT, Physics 2013 Fall 1 Introduction 2 The Gibbs Factor Gibbs Factor Several examples 3 Quantum Statistics From high T to low T From Particle States to Occupation Numbers

More information

Thermodynamics, Gibbs Method and Statistical Physics of Electron Gases

Thermodynamics, Gibbs Method and Statistical Physics of Electron Gases Bahram M. Askerov Sophia R. Figarova Thermodynamics, Gibbs Method and Statistical Physics of Electron Gases With im Figures Springer Contents 1 Basic Concepts of Thermodynamics and Statistical Physics...

More information

PHYS 352 Homework 2 Solutions

PHYS 352 Homework 2 Solutions PHYS 352 Homework 2 Solutions Aaron Mowitz (, 2, and 3) and Nachi Stern (4 and 5) Problem The purpose of doing a Legendre transform is to change a function of one or more variables into a function of variables

More information

Part II Statistical Physics

Part II Statistical Physics Part II Statistical Physics Theorems Based on lectures by H. S. Reall Notes taken by Dexter Chua Lent 2017 These notes are not endorsed by the lecturers, and I have modified them (often significantly)

More information

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

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 =

More information

Thermal & Statistical Physics Study Questions for the Spring 2018 Department Exam December 6, 2017

Thermal & Statistical Physics Study Questions for the Spring 2018 Department Exam December 6, 2017 Thermal & Statistical Physics Study Questions for the Spring 018 Department Exam December 6, 017 1. a. Define the chemical potential. Show that two systems are in diffusive equilibrium if 1. You may start

More information

First Problem Set for Physics 847 (Statistical Physics II)

First Problem Set for Physics 847 (Statistical Physics II) First Problem Set for Physics 847 (Statistical Physics II) Important dates: Feb 0 0:30am-:8pm midterm exam, Mar 6 9:30am-:8am final exam Due date: Tuesday, Jan 3. Review 0 points Let us start by reviewing

More information

(# = %(& )(* +,(- Closed system, well-defined energy (or e.g. E± E/2): Microcanonical ensemble

(# = %(& )(* +,(- Closed system, well-defined energy (or e.g. E± E/2): Microcanonical ensemble Recall from before: Internal energy (or Entropy): &, *, - (# = %(& )(* +,(- Closed system, well-defined energy (or e.g. E± E/2): Microcanonical ensemble & = /01Ω maximized Ω: fundamental statistical quantity

More information

Kinetic Theory 1 / Probabilities

Kinetic Theory 1 / Probabilities Kinetic Theory 1 / Probabilities 1. Motivations: statistical mechanics and fluctuations 2. Probabilities 3. Central limit theorem 1 Reading check Main concept introduced in first half of this chapter A)Temperature

More information

Introduction. Chapter The Purpose of Statistical Mechanics

Introduction. Chapter The Purpose of Statistical Mechanics Chapter 1 Introduction 1.1 The Purpose of Statistical Mechanics Statistical Mechanics is the mechanics developed to treat a collection of a large number of atoms or particles. Such a collection is, for

More information

summary of statistical physics

summary of statistical physics summary of statistical physics Matthias Pospiech University of Hannover, Germany Contents 1 Probability moments definitions 3 2 bases of thermodynamics 4 2.1 I. law of thermodynamics..........................

More information

Internal Degrees of Freedom

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

More information

Physics 112 The Classical Ideal Gas

Physics 112 The Classical Ideal Gas Physics 112 The Classical Ideal Gas Peter Young (Dated: February 6, 2012) We will obtain the equation of state and other properties, such as energy and entropy, of the classical ideal gas. We will start

More information

We already came across a form of indistinguishably in the canonical partition function: V N Q =

We already came across a form of indistinguishably in the canonical partition function: V N Q = Bosons en fermions Indistinguishability We already came across a form of indistinguishably in the canonical partition function: for distinguishable particles Q = Λ 3N βe p r, r 2,..., r N ))dτ dτ 2...

More information

Fermi-Dirac and Bose-Einstein

Fermi-Dirac and Bose-Einstein Fermi-Dirac and Bose-Einstein Beg. chap. 6 and chap. 7 of K &K Quantum Gases Fermions, Bosons Partition functions and distributions Density of states Non relativistic Relativistic Classical Limit Fermi

More information

Phys Midterm. March 17

Phys Midterm. March 17 Phys 7230 Midterm March 17 Consider a spin 1/2 particle fixed in space in the presence of magnetic field H he energy E of such a system can take one of the two values given by E s = µhs, where µ is the

More information

fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES

fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Content-Thermodynamics & Statistical Mechanics 1. Kinetic theory of gases..(1-13) 1.1 Basic assumption of kinetic theory 1.1.1 Pressure exerted by a gas 1.2 Gas Law for Ideal gases: 1.2.1 Boyle s Law 1.2.2

More information

Statistical Mechanics

Statistical Mechanics Statistical Mechanics Newton's laws in principle tell us how anything works But in a system with many particles, the actual computations can become complicated. We will therefore be happy to get some 'average'

More information

Introduction Statistical Thermodynamics. Monday, January 6, 14

Introduction Statistical Thermodynamics. Monday, January 6, 14 Introduction Statistical Thermodynamics 1 Molecular Simulations Molecular dynamics: solve equations of motion Monte Carlo: importance sampling r 1 r 2 r n MD MC r 1 r 2 2 r n 2 3 3 4 4 Questions How can

More information

(i) T, p, N Gibbs free energy G (ii) T, p, µ no thermodynamic potential, since T, p, µ are not independent of each other (iii) S, p, N Enthalpy H

(i) T, p, N Gibbs free energy G (ii) T, p, µ no thermodynamic potential, since T, p, µ are not independent of each other (iii) S, p, N Enthalpy H Solutions exam 2 roblem 1 a Which of those quantities defines a thermodynamic potential Why? 2 points i T, p, N Gibbs free energy G ii T, p, µ no thermodynamic potential, since T, p, µ are not independent

More information

Statistical Mechanics

Statistical Mechanics Franz Schwabl Statistical Mechanics Translated by William Brewer Second Edition With 202 Figures, 26 Tables, and 195 Problems 4u Springer Table of Contents 1. Basic Principles 1 1.1 Introduction 1 1.2

More information

Thermodynamics & Statistical Mechanics

Thermodynamics & Statistical Mechanics hysics GRE: hermodynamics & Statistical Mechanics G. J. Loges University of Rochester Dept. of hysics & Astronomy xkcd.com/66/ c Gregory Loges, 206 Contents Ensembles 2 Laws of hermodynamics 3 hermodynamic

More information

Physics Oct Reading. K&K chapter 6 and the first half of chapter 7 (the Fermi gas). The Ideal Gas Again

Physics Oct Reading. K&K chapter 6 and the first half of chapter 7 (the Fermi gas). The Ideal Gas Again Physics 301 11-Oct-004 14-1 Reading K&K chapter 6 and the first half of chapter 7 the Fermi gas) The Ideal Gas Again Using the grand partition function we ve discussed the Fermi-Dirac and Bose-Einstein

More information

The non-interacting Bose gas

The non-interacting Bose gas Chapter The non-interacting Bose gas Learning goals What is a Bose-Einstein condensate and why does it form? What determines the critical temperature and the condensate fraction? What changes for trapped

More information

6.730 Physics for Solid State Applications

6.730 Physics for Solid State Applications 6.730 Physics for Solid State Applications Lecture 25: Chemical Potential and Equilibrium Outline Microstates and Counting System and Reservoir Microstates Constants in Equilibrium Temperature & Chemical

More information

Monatomic ideal gas: partition functions and equation of state.

Monatomic ideal gas: partition functions and equation of state. Monatomic ideal gas: partition functions and equation of state. Peter Košovan peter.kosovan@natur.cuni.cz Dept. of Physical and Macromolecular Chemistry Statistical Thermodynamics, MC260P105, Lecture 3,

More information

PHYS3113, 3d year Statistical Mechanics Tutorial problems. Tutorial 1, Microcanonical, Canonical and Grand Canonical Distributions

PHYS3113, 3d year Statistical Mechanics Tutorial problems. Tutorial 1, Microcanonical, Canonical and Grand Canonical Distributions 1 PHYS3113, 3d year Statistical Mechanics Tutorial problems Tutorial 1, Microcanonical, Canonical and Grand Canonical Distributions Problem 1 The macrostate probability in an ensemble of N spins 1/2 is

More information

Chemical Potential (a Summary)

Chemical Potential (a Summary) Chemical Potential a Summary Definition and interpretations KK chap 5. Thermodynamics definition Concentration Normalization Potential Law of mass action KK chap 9 Saha Equation The density of baryons

More information

( ) ( )( k B ( ) ( ) ( ) ( ) ( ) ( k T B ) 2 = ε F. ( ) π 2. ( ) 1+ π 2. ( k T B ) 2 = 2 3 Nε 1+ π 2. ( ) = ε /( ε 0 ).

( ) ( )( k B ( ) ( ) ( ) ( ) ( ) ( k T B ) 2 = ε F. ( ) π 2. ( ) 1+ π 2. ( k T B ) 2 = 2 3 Nε 1+ π 2. ( ) = ε /( ε 0 ). PHYS47-Statistical Mechanics and hermal Physics all 7 Assignment #5 Due on November, 7 Problem ) Problem 4 Chapter 9 points) his problem consider a system with density of state D / ) A) o find the ermi

More information

Ideal gases. Asaf Pe er Classical ideal gas

Ideal gases. Asaf Pe er Classical ideal gas Ideal gases Asaf Pe er 1 November 2, 213 1. Classical ideal gas A classical gas is generally referred to as a gas in which its molecules move freely in space; namely, the mean separation between the molecules

More information

Phonon II Thermal Properties

Phonon II Thermal Properties Phonon II Thermal Properties Physics, UCF OUTLINES Phonon heat capacity Planck distribution Normal mode enumeration Density of states in one dimension Density of states in three dimension Debye Model for

More information

Please read the following instructions:

Please read the following instructions: MIDTERM #1 PHYS 33 (MODERN PHYSICS II) DATE/TIME: February 16, 17 (8:3 a.m. - 9:45 a.m.) PLACE: RB 11 Only non-programmable calculators are allowed. Name: ID: Please read the following instructions: This

More information

Supplement: Statistical Physics

Supplement: Statistical Physics Supplement: Statistical Physics Fitting in a Box. Counting momentum states with momentum q and de Broglie wavelength λ = h q = 2π h q In a discrete volume L 3 there is a discrete set of states that satisfy

More information

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:- 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

More information

Kinetic Theory 1 / Probabilities

Kinetic Theory 1 / Probabilities Kinetic Theory 1 / Probabilities 1. Motivations: statistical mechanics and fluctuations 2. Probabilities 3. Central limit theorem 1 The need for statistical mechanics 2 How to describe large systems In

More information

In-class exercises. Day 1

In-class exercises. Day 1 Physics 4488/6562: Statistical Mechanics http://www.physics.cornell.edu/sethna/teaching/562/ Material for Week 8 Exercises due Mon March 19 Last correction at March 5, 2018, 8:48 am c 2017, James Sethna,

More information

Physics 607 Final Exam

Physics 607 Final Exam Physics 67 Final Exam Please be well-organized, and show all significant steps clearly in all problems. You are graded on your work, so please do not just write down answers with no explanation! Do all

More information

Chapter 2: Equation of State

Chapter 2: Equation of State Introduction Chapter 2: Equation of State The Local Thermodynamic Equilibrium The Distribution Function Black Body Radiation Fermi-Dirac EoS The Complete Degenerate Gas Application to White Dwarfs Temperature

More information

Noninteracting Particle Systems

Noninteracting Particle Systems Chapter 6 Noninteracting Particle Systems c 26 by Harvey Gould and Jan Tobochnik 8 December 26 We apply the general formalism of statistical mechanics to classical and quantum systems of noninteracting

More information

Imperial College London BSc/MSci EXAMINATION May 2008 THERMODYNAMICS & STATISTICAL PHYSICS

Imperial College London BSc/MSci EXAMINATION May 2008 THERMODYNAMICS & STATISTICAL PHYSICS Imperial College London BSc/MSci EXAMINATION May 2008 This paper is also taken for the relevant Examination for the Associateship THERMODYNAMICS & STATISTICAL PHYSICS For Second-Year Physics Students Wednesday,

More information

Physics 2203, Fall 2011 Modern Physics

Physics 2203, Fall 2011 Modern Physics Physics 2203, Fall 2011 Modern Physics. Friday, Nov. 2 nd, 2012. Energy levels in Nitrogen molecule Sta@s@cal Physics: Quantum sta@s@cs: Ch. 15 in our book. Notes from Ch. 10 in Serway Announcements Second

More information

Summary Part Thermodynamic laws Thermodynamic processes. Fys2160,

Summary Part Thermodynamic laws Thermodynamic processes. Fys2160, ! Summary Part 2 21.11.2018 Thermodynamic laws Thermodynamic processes Fys2160, 2018 1 1 U is fixed ) *,,, -(/,,), *,, -(/,,) N, 3 *,, - /,,, 2(3) Summary Part 1 Equilibrium statistical systems CONTINUE...

More information

Advanced Thermodynamics. Jussi Eloranta (Updated: January 22, 2018)

Advanced Thermodynamics. Jussi Eloranta (Updated: January 22, 2018) Advanced Thermodynamics Jussi Eloranta (jmeloranta@gmail.com) (Updated: January 22, 2018) Chapter 1: The machinery of statistical thermodynamics A statistical model that can be derived exactly from the

More information

Grand Canonical Formalism

Grand Canonical Formalism Grand Canonical Formalism Grand Canonical Ensebmle For the gases of ideal Bosons and Fermions each single-particle mode behaves almost like an independent subsystem, with the only reservation that the

More information

Physics 4230 Final Examination 10 May 2007

Physics 4230 Final Examination 10 May 2007 Physics 43 Final Examination May 7 In each problem, be sure to give the reasoning for your answer and define any variables you create. If you use a general formula, state that formula clearly before manipulating

More information

A Brief Introduction to Statistical Mechanics

A Brief Introduction to Statistical Mechanics A Brief Introduction to Statistical Mechanics E. J. Maginn, J. K. Shah Department of Chemical and Biomolecular Engineering University of Notre Dame Notre Dame, IN 46556 USA Monte Carlo Workshop Universidade

More information

Many-Particle Systems

Many-Particle Systems Chapter 6 Many-Particle Systems c 21 by Harvey Gould and Jan Tobochnik 2 December 21 We apply the general formalism of statistical mechanics to systems of many particles and discuss the semiclassical limit

More information

9.1 System in contact with a heat reservoir

9.1 System in contact with a heat reservoir Chapter 9 Canonical ensemble 9. System in contact with a heat reservoir We consider a small system A characterized by E, V and N in thermal interaction with a heat reservoir A 2 characterized by E 2, V

More information

Please read the following instructions:

Please read the following instructions: MIDTERM #1 PHYS 33 (MODERN PHYSICS II) DATE/TIME: February 16, 17 (8:3 a.m. - 9:45 a.m.) PLACE: RB 11 Only non-programmable calculators are allowed. Name: ID: Please read the following instructions: This

More information

The Ideal Gas. One particle in a box:

The Ideal Gas. One particle in a box: IDEAL GAS The Ideal Gas It is an important physical example that can be solved exactly. All real gases behave like ideal if the density is small enough. In order to derive the law, we have to do following:

More information

PAPER No. 6: PHYSICAL CHEMISTRY-II (Statistical

PAPER No. 6: PHYSICAL CHEMISTRY-II (Statistical Subject Paper No and Title Module No and Title Module Tag 6: PHYSICAL CHEMISTRY-II (Statistical 1: Introduction to Statistical CHE_P6_M1 TABLE OF CONTENTS 1. Learning Outcomes 2. Statistical Mechanics

More information

Unit III Free Electron Theory Engineering Physics

Unit III Free Electron Theory Engineering Physics . Introduction The electron theory of metals aims to explain the structure and properties of solids through their electronic structure. The electron theory is applicable to all solids i.e., both metals

More information

Microcanonical Ensemble

Microcanonical Ensemble Entropy for Department of Physics, Chungbuk National University October 4, 2018 Entropy for A measure for the lack of information (ignorance): s i = log P i = log 1 P i. An average ignorance: S = k B i

More information

MD Thermodynamics. Lecture 12 3/26/18. Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky

MD Thermodynamics. Lecture 12 3/26/18. Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky MD Thermodynamics Lecture 1 3/6/18 1 Molecular dynamics The force depends on positions only (not velocities) Total energy is conserved (micro canonical evolution) Newton s equations of motion (second order

More information

Statistical Mechanics Notes. Ryan D. Reece

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..........................

More information

Physics 408 Final Exam

Physics 408 Final Exam Physics 408 Final Exam Name You are graded on your work, with partial credit where it is deserved. Please give clear, well-organized solutions. 1. Consider the coexistence curve separating two different

More information

Physics 607 Final Exam

Physics 607 Final Exam Physics 607 Final Exam Please be well-organized, and show all significant steps clearly in all problems. You are graded on your work, so please do not just write down answers with no explanation! Do all

More information

Part II: Statistical Physics

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

More information

Statistical Mechanics I

Statistical Mechanics I Statistical Mechanics I Peter S. Riseborough November 15, 11 Contents 1 Introduction 4 Thermodynamics 4.1 The Foundations of Thermodynamics............... 4. Thermodynamic Equilibrium....................

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Statistical Physics I Spring Term Solutions to Problem Set #10

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Statistical Physics I Spring Term Solutions to Problem Set #10 MASSACHUSES INSIUE OF ECHNOLOGY Physics Department 8.044 Statistical Physics I Spring erm 203 Problem : wo Identical Particles Solutions to Problem Set #0 a) Fermions:,, 0 > ɛ 2 0 state, 0, > ɛ 3 0,, >

More information

1 Introduction. 1.1 The Subject Matter of Statistical Mechanics. Lies, damned lies, and statistics. Disraeli

1 Introduction. 1.1 The Subject Matter of Statistical Mechanics. Lies, damned lies, and statistics. Disraeli 1 Introduction Lies, damned lies, and statistics. Disraeli 1.1 The Subject Matter of Statistical Mechanics The goal of statistical mechanics is to predict the macroscopic properties of bodies, most especially

More information

Lecture Notes, Statistical Mechanics (Theory F) Jörg Schmalian

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

More information

Statistical Mechanics Victor Naden Robinson vlnr500 3 rd Year MPhys 17/2/12 Lectured by Rex Godby

Statistical Mechanics Victor Naden Robinson vlnr500 3 rd Year MPhys 17/2/12 Lectured by Rex Godby Statistical Mechanics Victor Naden Robinson vlnr500 3 rd Year MPhys 17/2/12 Lectured by Rex Godby Lecture 1: Probabilities Lecture 2: Microstates for system of N harmonic oscillators Lecture 3: More Thermodynamics,

More information

Physics 404: Final Exam Name (print): "I pledge on my honor that I have not given or received any unauthorized assistance on this examination.

Physics 404: Final Exam Name (print): I pledge on my honor that I have not given or received any unauthorized assistance on this examination. Physics 404: Final Exam Name (print): "I pledge on my honor that I have not given or received any unauthorized assistance on this examination." May 20, 2008 Sign Honor Pledge: Don't get bogged down on

More information

Statistical Thermodynamics and Monte-Carlo Evgenii B. Rudnyi and Jan G. Korvink IMTEK Albert Ludwig University Freiburg, Germany

Statistical Thermodynamics and Monte-Carlo Evgenii B. Rudnyi and Jan G. Korvink IMTEK Albert Ludwig University Freiburg, Germany Statistical Thermodynamics and Monte-Carlo Evgenii B. Rudnyi and Jan G. Korvink IMTEK Albert Ludwig University Freiburg, Germany Preliminaries Learning Goals From Micro to Macro Statistical Mechanics (Statistical

More information

Definite Integral and the Gibbs Paradox

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,

More information

Part II: Statistical Physics

Part II: Statistical Physics Chapter 6: Boltzmann Statistics SDSMT, Physics Fall Semester: Oct. - Dec., 2013 1 Introduction: Very brief 2 Boltzmann Factor Isolated System and System of Interest Boltzmann Factor The Partition Function

More information

Elements of Statistical Mechanics

Elements of Statistical Mechanics Elements of Statistical Mechanics Thermodynamics describes the properties of macroscopic bodies. Statistical mechanics allows us to obtain the laws of thermodynamics from the laws of mechanics, classical

More information

13. Ideal Quantum Gases I: Bosons

13. Ideal Quantum Gases I: Bosons University of Rhode Island DigitalCommons@URI Equilibrium Statistical Physics Physics Course Materials 5 3. Ideal Quantum Gases I: Bosons Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative

More information

424 Index. Eigenvalue in quantum mechanics, 174 eigenvector in quantum mechanics, 174 Einstein equation, 334, 342, 393

424 Index. Eigenvalue in quantum mechanics, 174 eigenvector in quantum mechanics, 174 Einstein equation, 334, 342, 393 Index After-effect function, 368, 369 anthropic principle, 232 assumptions nature of, 242 autocorrelation function, 292 average, 18 definition of, 17 ensemble, see ensemble average ideal,23 operational,

More information

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 =

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

More information

The perfect quantal gas

The perfect quantal gas The perfect quantal gas Asaf Pe er 1 March 27, 2013 1. Background So far in this course we have been discussing ideal classical gases. We saw that the conditions for gases to be treated classically are

More information

PHY 6500 Thermal and Statistical Physics - Fall 2017

PHY 6500 Thermal and Statistical Physics - Fall 2017 PHY 6500 Thermal and Statistical Physics - Fall 2017 Time: M, F 12:30 PM 2:10 PM. From 08/30/17 to 12/19/17 Place: Room 185 Physics Research Building Lecturer: Boris Nadgorny E-mail: nadgorny@physics.wayne.edu

More information

Statistical thermodynamics L1-L3. Lectures 11, 12, 13 of CY101

Statistical thermodynamics L1-L3. Lectures 11, 12, 13 of CY101 Statistical thermodynamics L1-L3 Lectures 11, 12, 13 of CY101 Need for statistical thermodynamics Microscopic and macroscopic world Distribution of energy - population Principle of equal a priori probabilities

More information

Appendix 1: Normal Modes, Phase Space and Statistical Physics

Appendix 1: Normal Modes, Phase Space and Statistical Physics Appendix : Normal Modes, Phase Space and Statistical Physics The last line of the introduction to the first edition states that it is the wide validity of relatively few principles which this book seeks

More information

ChE 210B: Advanced Topics in Equilibrium Statistical Mechanics

ChE 210B: Advanced Topics in Equilibrium Statistical Mechanics ChE 210B: Advanced Topics in Equilibrium Statistical Mechanics Glenn Fredrickson Lecture 1 Reading: 3.1-3.5 Chandler, Chapters 1 and 2 McQuarrie This course builds on the elementary concepts of statistical

More information

Classical Statistical Mechanics: Part 1

Classical Statistical Mechanics: Part 1 Classical Statistical Mechanics: Part 1 January 16, 2013 Classical Mechanics 1-Dimensional system with 1 particle of mass m Newton s equations of motion for position x(t) and momentum p(t): ẋ(t) dx p =

More information

Physics 576 Stellar Astrophysics Prof. James Buckley. Lecture 14 Relativistic Quantum Mechanics and Quantum Statistics

Physics 576 Stellar Astrophysics Prof. James Buckley. Lecture 14 Relativistic Quantum Mechanics and Quantum Statistics Physics 576 Stellar Astrophysics Prof. James Buckley Lecture 14 Relativistic Quantum Mechanics and Quantum Statistics Reading/Homework Assignment Read chapter 3 in Rose. Midterm Exam, April 5 (take home)

More information

Table of Contents [ttc]

Table of Contents [ttc] Table of Contents [ttc] 1. Equilibrium Thermodynamics I: Introduction Thermodynamics overview. [tln2] Preliminary list of state variables. [tln1] Physical constants. [tsl47] Equations of state. [tln78]

More information

Physics 607 Exam 2. ( ) = 1, Γ( z +1) = zγ( z) x n e x2 dx = 1. e x2

Physics 607 Exam 2. ( ) = 1, Γ( z +1) = zγ( z) x n e x2 dx = 1. e x2 Physics 607 Exam Please be well-organized, and show all significant steps clearly in all problems. You are graded on your work, so please do not just write down answers with no explanation! Do all your

More information

2. Thermodynamics. Introduction. Understanding Molecular Simulation

2. Thermodynamics. Introduction. Understanding Molecular Simulation 2. Thermodynamics Introduction Molecular Simulations Molecular dynamics: solve equations of motion r 1 r 2 r n Monte Carlo: importance sampling r 1 r 2 r n How do we know our simulation is correct? Molecular

More information

PRINCIPLES OF PHYSICS. \Hp. Ni Jun TSINGHUA. Physics. From Quantum Field Theory. to Classical Mechanics. World Scientific. Vol.2. Report and Review in

PRINCIPLES OF PHYSICS. \Hp. Ni Jun TSINGHUA. Physics. From Quantum Field Theory. to Classical Mechanics. World Scientific. Vol.2. Report and Review in LONDON BEIJING HONG TSINGHUA Report and Review in Physics Vol2 PRINCIPLES OF PHYSICS From Quantum Field Theory to Classical Mechanics Ni Jun Tsinghua University, China NEW JERSEY \Hp SINGAPORE World Scientific

More information

Contents. 1 Introduction and guide for this text 1. 2 Equilibrium and entropy 6. 3 Energy and how the microscopic world works 21

Contents. 1 Introduction and guide for this text 1. 2 Equilibrium and entropy 6. 3 Energy and how the microscopic world works 21 Preface Reference tables Table A Counting and combinatorics formulae Table B Useful integrals, expansions, and approximations Table C Extensive thermodynamic potentials Table D Intensive per-particle thermodynamic

More information

Non-Continuum Energy Transfer: Phonons

Non-Continuum Energy Transfer: Phonons Non-Continuum Energy Transfer: Phonons D. B. Go Slide 1 The Crystal Lattice The crystal lattice is the organization of atoms and/or molecules in a solid simple cubic body-centered cubic hexagonal a NaCl

More information

Bohr s Model, Energy Bands, Electrons and Holes

Bohr s Model, Energy Bands, Electrons and Holes Dual Character of Material Particles Experimental physics before 1900 demonstrated that most of the physical phenomena can be explained by Newton's equation of motion of material particles or bodies and

More information

Solutions to Problem Set 8

Solutions to Problem Set 8 Cornell University, Physics Department Fall 2014 PHYS-3341 Statistical Physics Prof. Itai Cohen Solutions to Problem Set 8 David C. sang, Woosong Choi 8.1 Chemical Equilibrium Reif 8.12: At a fixed temperature

More information

CHAPTER 9 Statistical Physics

CHAPTER 9 Statistical Physics CHAPTER 9 Statistical Physics 9.1 9.2 9.3 9.4 9.5 9.6 9.7 Historical Overview Maxwell Velocity Distribution Equipartition Theorem Maxwell Speed Distribution Classical and Quantum Statistics Fermi-Dirac

More information

Statistical. mechanics

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

More information

Statistical thermodynamics for MD and MC simulations

Statistical thermodynamics for MD and MC simulations Statistical thermodynamics for MD and MC simulations knowing 2 atoms and wishing to know 10 23 of them Marcus Elstner and Tomáš Kubař 22 June 2016 Introduction Thermodynamic properties of molecular systems

More information

1. Thermodynamics 1.1. A macroscopic view of matter

1. Thermodynamics 1.1. A macroscopic view of matter 1. Thermodynamics 1.1. A macroscopic view of matter Intensive: independent of the amount of substance, e.g. temperature,pressure. Extensive: depends on the amount of substance, e.g. internal energy, enthalpy.

More information

Homework Hint. Last Time

Homework Hint. Last Time Homework Hint Problem 3.3 Geometric series: ωs τ ħ e s= 0 =? a n ar = For 0< r < 1 n= 0 1 r ωs τ ħ e s= 0 1 = 1 e ħω τ Last Time Boltzmann factor Partition Function Heat Capacity The magic of the partition

More information

Identical Particles. Bosons and Fermions

Identical Particles. Bosons and Fermions Identical Particles Bosons and Fermions In Quantum Mechanics there is no difference between particles and fields. The objects which we refer to as fields in classical physics (electromagnetic field, field

More information

Kinetic theory of the ideal gas

Kinetic theory of the ideal gas Appendix H Kinetic theory of the ideal gas This Appendix contains sketchy notes, summarizing the main results of elementary kinetic theory. The students who are not familiar with these topics should refer

More information

Lecture 6: Ideal gas ensembles

Lecture 6: Ideal gas ensembles Introduction Lecture 6: Ideal gas ensembles A simple, instructive and practical application of the equilibrium ensemble formalisms of the previous lecture concerns an ideal gas. Such a physical system

More information

ESE 372 / Spring 2013 / Lecture 5 Metal Oxide Semiconductor Field Effect Transistor

ESE 372 / Spring 2013 / Lecture 5 Metal Oxide Semiconductor Field Effect Transistor Metal Oxide Semiconductor Field Effect Transistor V G V G 1 Metal Oxide Semiconductor Field Effect Transistor We will need to understand how this current flows through Si What is electric current? 2 Back

More information

Suggestions for Further Reading

Suggestions for Further Reading Contents Preface viii 1 From Microscopic to Macroscopic Behavior 1 1.1 Introduction........................................ 1 1.2 Some Qualitative Observations............................. 2 1.3 Doing

More information