Lecture 10Lecture 10——Ideas of Statistical Mechanics Ideas of Statistical Mechanics Chapter 4, Chapter 4, Wednesday January 30Wednesday January 30thth•Finish Ch. 3 - Statistical distributions•Statistical mechanics - ideas and definitions•Quantum states, classical probability, ensembles, macrostates...•Entropy•Definition of a quantum stateReading: Reading: All of chapter 4 (pages 67 All of chapter 4 (pages 67 --88)88)***Homework 3 due Fri. Feb. 1st*******Homework 3 due Fri. Feb. 1st****Assigned problems, Assigned problems, Ch. 3Ch. 3: 8, 10, 16, 18, 20: 8, 10, 16, 18, 20Homework 4 due next Thu. Feb. 7thHomework 4 due next Thu. Feb. 7thAssigned problems, Assigned problems, Ch. 4Ch. 4: 2, 8, 10, 12, 14: 2, 8, 10, 12, 14Exam 1: Exam 1: Fri. Feb. 8th (in class), chapters 1Fri. Feb. 8th (in class), chapters 1--44Statistical distributionsStatistical distributionsnixi16,whereiiiiinxxNnN==∑∑Mean:Statistical distributionsStatistical distributionsnixi16,whereiiiiinxxNnN==∑∑Mean:Statistical distributionsStatistical distributionsnixi16,where limiii iiNnxpx pN→∞==∑Mean:N →∞Statistical distributionsStatistical distributionsnixi16() ()22iiixpxxσ=Δ = −∑Standard deviation()221() exp22xxpxσσπ⎧⎫−⎪⎪=−⎨⎬⎪⎪⎩⎭Statistical distributionsStatistical distributionsGaussian distribution(Bell curve)64Statistical Mechanics (Chapter 4)Statistical Mechanics (Chapter 4)••What is the physical basis for the 2nd law?What is the physical basis for the 2nd law?••What is the microscopic basis for entropy?What is the microscopic basis for entropy?Boltzmann hypothesis: the entropy of a system is related to the probability of its state; the basis of entropy is statistical.Statistics + MechanicsStatistics + MechanicsStatistical MechanicsStatistical MechanicsThermal PropertiesThermal PropertiesStatistical MechanicsStatistical Mechanics••Use classical probability to make predictions.Use classical probability to make predictions.••Use statistical probability to test predictions.Use statistical probability to test predictions.Note: statistical probability has no basis if a system is out ofequilibrium (repeat tests, get different results).How on earth is this possible?How on earth is this possible?••How do we define simple events?How do we define simple events?••How do we count them?How do we count them?••How can we be sure they have equal probabilities?How can we be sure they have equal probabilities?REQUIRES AN IMMENSE LEAP OF FAITHREQUIRES AN IMMENSE LEAP OF FAITHStatistical Mechanics Statistical Mechanics ––ideas and definitionsideas and definitionsA quantum state, or microstateA quantum state, or microstate••A unique configuration.A unique configuration.••To know that it is unique, we must specify it as To know that it is unique, we must specify it as completely as possible...completely as possible...e.g. Determine:e.g. Determine:PositionPositionMomentumMomentumEnergyEnergySpinSpinof every particle, all at once!!!!!of every particle, all at once!!!!!............THIS IS ACTUALLY IMPOSSIBLE FOR ANY REAL SYSTEMTHIS IS ACTUALLY IMPOSSIBLE FOR ANY REAL SYSTEMStatistical Mechanics Statistical Mechanics ––ideas and definitionsideas and definitionsA quantum state, or microstateA quantum state, or microstate••A unique configuration.A unique configuration.••To know that it is unique, we must specify it as To know that it is unique, we must specify it as completely as possible...completely as possible...Classical probabilityClassical probability••Cannot use statistical probability.Cannot use statistical probability.••Thus, we are forced to use classical probability.Thus, we are forced to use classical probability.An ensembleAn ensemble••A collection of separate systems prepared in A collection of separate systems prepared in precisely the same way.precisely the same way.Statistical Mechanics Statistical Mechanics ––ideas and definitionsideas and definitionsThe The microcanonicalmicrocanonicalensemble:ensemble:Each system has same:Each system has same:# of particles# of particlesTotal energyTotal energyVolumeVolumeShapeShapeMagnetic fieldMagnetic fieldElectric fieldElectric fieldand so on....and so on................These variables (parameters) specify the These variables (parameters) specify the ‘‘macrostatemacrostate’’of the ensemble. A of the ensemble. A macrostatemacrostateis specified by is specified by ‘‘an an equation of stateequation of state’’. Many, many different microstates . Many, many different microstates might correspond to the same might correspond to the same macrostatemacrostate..64Statistical Mechanics Statistical Mechanics ––ideas and definitionsideas and definitionsAn example:An example:Coin toss again!!widthEnsembles and quantum states (microstates)Ensembles and quantum states (microstates)Cell volume, Cell volume, ΔΔVVVolume Volume VV10 particles, 36 cells10 particles, 36 cells1016136310ip−⎛⎞=⎜⎟⎝⎠≈×Ensembles and quantum states (microstates)Ensembles and quantum states (microstates)Cell volume, Cell volume, ΔΔVVVolume Volume VV10 particles, 36 cells10 particles, 36 cells1016136310ip−⎛⎞=⎜⎟⎝⎠≈×Ensembles and quantum states (microstates)Ensembles and quantum states (microstates)Cell volume, Cell volume, ΔΔVVVolume Volume VV10 particles, 36 cells10 particles, 36 cells1016136310ip−⎛⎞=⎜⎟⎝⎠≈×Ensembles and quantum states (microstates)Ensembles and quantum states (microstates)Cell volume, Cell volume, ΔΔVVVolume Volume VV10 particles, 36 cells10 particles, 36 cells1016136310ip−⎛⎞=⎜⎟⎝⎠≈×Ensembles and quantum states (microstates)Ensembles and quantum states (microstates)Cell volume, Cell volume, ΔΔVVVolume Volume VV10 particles, 36 cells10 particles, 36 cells1016136310ip−⎛⎞=⎜⎟⎝⎠≈×Ensembles and quantum states (microstates)Ensembles and quantum states (microstates)Cell volume, Cell volume, ΔΔVVVolume Volume VV10 particles, 36 cells10 particles, 36 cells1016136310ip−⎛⎞=⎜⎟⎝⎠≈×Ensembles and quantum states (microstates)Ensembles and quantum states (microstates)Cell volume, Cell volume, ΔΔVVVolume Volume VV10 particles, 36 cells10 particles, 36 cells1016136310ip−⎛⎞=⎜⎟⎝⎠≈×Ensembles and quantum states (microstates)Ensembles and quantum states (microstates)Cell volume, Cell volume, ΔΔVVVolume Volume VV10 particles, 36 cells10 particles, 36