Statistical Physics How to connect the microscopic properties -- lots of changes to the macroscopic properties -- not changing much. We will care about: N = # atoms T = temperature V = volume U = total energy Deep philosophical issue: Microscopic: physical laws are reversible in time Macroscopic: They re not always reversible! Reversibility Movie taken and modified from: www.theclam.com/video/jumpingindex.asp 1
The Fundamental Assumptions of Statistical Physics 1. Quantum States are either accessible or inaccessible to the system IN The System OUT 2. The System is equally likely to be in any accessible state. Last Time - Sharpness of Multiplicity Function Stirling s Approximation Gaussian Distribution ( ) 1 2 π N N! 2 N N e N -Binary (two state) model multiplicity function approaches Gaussian at large N: -RMS fluctuations ~ -Weighted averages 2
Today Chapter 2 Entropy and Temperature Fundamental assumption Closed System Probability and Ensembles Two systems in thermal contact Entropy Temperature Laws of thermodynamics Fundamental Assumption Closed system: clueless of outside world constant energy (δu/u << 1) constant number of particles (δn/n << 1) constant volume Closed system is equally likely to be in any accessible state Accessible state: compatible with macrostate i.e. at the right energy, number of particles, volume Note: some states can be inaccessible due to time constraints. Diamond and graphite can one spontaneously convert into the other? 3
Probability System is equally likely to be in any accessible state. Multiplicity = g = number of accessible states ~ 10 23 Probability of any one state: All microstates are equally likely. s = general state label. Always Normalize: Ensemble ( 1 + 1 ) ( 2 + 2 ) = 1 2 + 1 2 + 1 2 + 1 2 System can be in any accessible state Ensemble a set of systems that are all alike. - one copy of the real system for each accessible state Multiplicity = g = number of accessible states There are g states in the ensemble. To do math, replace The System with The Ensemble 4
Ensemble Average (Last time, we called this a weighted average) < > Denotes an average X is some property, like magnetization Average over the ensemble of accessible states ( 1 + 1 ) ( 2 + 2 ) = 1 2 + 1 2 + 1 2 + 1 2 Example of an Ensemble: Spins The System: 6 spins magnetization microstate 6 2 2 4 (Only 4 states accessible due to some constraints) Average over this ensemble: (6+2+2+4)/4 = 3.5 The real ensemble has 2 6 = 64 microstates what is its spin average? 5
The System Has a particular energy U = energy U, N Contains a certain number of particles N = number of particles ~ 10 23 Has a multiplicity of microstates g = g(u,n) = number of microstates = number of configurations Two Systems in Thermal Contact Not in U 1 thermal U 2 N 1 contact N 2 thermally conducting barrier insulating barriers U 1 N 1 U 2 N 2 Exchange energy U, but not particles. Thermal contact 6
Two Systems in Thermal Contact U U 1 N 1 2 N 2 g 1 (U 1,N 1 ) g 2 (U 2,N 2 ) U 1 and U 2 will adjust. But how? Fundamental assumption: any microstate is equally possible The most probable macro state is the one with largest g Multiplicity: g( U 1,U 2 ) = g 1 (U 1,N 1 ) * g 2 (U 2,N 2 ) Trade energy δu: g( U 1 + δu, U 2 - δu) = g 1 (U 1 + δu,n 1 ) * g 2 (U 2 - δu,n 2 ) Need to find which energy gives the largest g Two Systems in Thermal Contact U U 1 N 1 2 N 2 g 1 (U 1,N 1 ) g 2 (U 2,N 2 ) U 1 + U 2 = U N 1 + N 2 = N Total multiplicity: Most Probable Configuration is the one for which g(u 1, N 1 )*g(u 2, N 2 ) is the largest. 7
Two spin system in magnetic field Reminder: Spin excess: s N N 2s B ½N N N U = mi B = mi B = 2smB i= 1 i= 1 - if s is positive for spins along magnetic field Zeeman splitting - states with different s split in energy scale, (degeneracy is removed) States with only one s are accessible at fixed energy! Two Spin Systems in Thermal Contact s 1 s 2 Total energy of the two systems combined: U U = 2s mb 1 1 = 2s mb 2 2 Exchange energy holding U constant Total multiplicity: Notice this is a function of the energy U(s) in the system. 8
Two Spin Systems in Thermal Contact s 1 s 2 Most Probable Configuration is the largest term in the sum: We will use and to denote the most probable spin states. Note that Largest term in sum: Multiplicity function is sharp average of the physical quantity over all states may be replaced by that quantity for that state only Entropy thermally conducting barrier They can trade energy U 1 N 1 g(u 1, N 1 ) U 2 N 2 g(u 2, N 2 ) Total multiplicity g(u 1, N 1 )* g(u 2, N 2 ) g = g g 1 2 Weird to multiply g*g to get total multiplicity. We d rather add two things to get total, like U = U 1 + U 2 and N = N 1 + N 2. 9
Entropy thermally conducting barrier They can trade energy U 1 N 1 g(u 1, N 1 ) U 2 N 2 g(u 2, N 2 ) Define the logarithm of g: Total multiplicity g(u 1, N 1 )* g(u 2, N 2 ) g = g g log 1 2 ( g ) = log ( g1 g2 ) ( g ) = ( g ) + ( g ) log log log Total entropy 1 2 ENTROPY Now we can add entropies! Thermal Equilibrium When two systems come into thermal contact, they will exchange energy (and adjust g 1 and g 2 ). The most likely state to find them in is the most probable configuration, which has the largest g 1 *g 2 THIS MEANS TOTAL ENTROPY IS MAXIMIZED. 10
Maximize Entropy In thermal equilibrium, Defines Temperature! Temperature When two systems are in thermal contact, they exchange energy until this condition is satisfied: And we know that in thermal equilibrium, the temperatures become the same: The two statements are equivalent. 11
Temperature T 1 is some function of and likewise for T 2. In KELVIN: Definition of Temperature k B = 1.381 x 10-23 Joules/Kelvin BOLTZMANN S CONSTANT Partial Derivatives You know: We really mean: while keeping a constant 12
Temperature Take Two thermally conducting barrier They can trade energy but not particles U 1 N 1 σ(u 1, N 1 ) U 2 N 2 σ(u 2, N 2 ) We really mean: Keeping number of particles constant on both sides. Temperature Take Two We really mean: 13
Temperature Take Two What we will use: Temperature in units of Energy Yes, you can just invert partial derivatives Two Spin Systems in Thermal Contact s 1 s 2 Most Probable Configuration is the largest term in the sum: We will use and to denote the most probable spin states. Note that Largest term in sum: In zero magnetic field, this becomes the binary model. 14
Example: Binary System in thermal contact s 1 s 2 N! g = g ( 0) e N! N! 2 2s N g 2 g ( 0) 2 N π N 1 2 s=0 s Example: Binary System in thermal contact At maximum, 1st derivative is zero, and 2nd derivative is negative. Easiest to work in terms of the log -- max happens at the same place. First derivative: 15
Example: Binary System in thermal contact At maximum, 1st derivative is zero, and 2nd derivative is negative. Easiest to work in terms of the log -- max happens at the same place. Second derivative: - It is always positive Example: Binary System in thermal contact s 1 s 2 Maximum configurations occurs for: or: (hat means at maximum ) fractional spin excess is the same for all systems in thermal contact! 16
Example: Sharpness of Multiplicity The most probable configuration is: s 1 s 2 Define deviations from the most probable values: Example: Sharpness of Multiplicity s 1 s 2 And the result is: this is the deviation from equilibrium, and it s small. If N 1 = N 2 = 10 22, and δ = 10 12, 2 δ 2 /N 1 = 200, and g 1 g 2 ~ e -400 ~ 10-174 below the maximum value (g 1 g 2 ) max 17
Laws of Thermodynamics Study Thermodynamics : Laws are Postulates Study Statistical Mechanics : Laws follow from the Fundamental Assumption Zeroth Law of Thermodynamics If A is in thermal equilibrium with B, and if B is in thermal equilibrium with C, A and C are in thermal equilibrium. If T A = T B and T B = T C, then T A = T C trade energy trade energy A B C 18
First Law of Thermodynamics Heat is a Form of Energy The (non-mechanical) flow of Energy from one system to another is Heat. (We ll make this more precise in chapter 8) This is how we keep track of the conservation of energy in statistical and thermal physics. We conserved energy between two systems in thermal contact, by letting them trade energy with each other, but not with the rest of the world. Second Law of Thermodynamics (Law of increase of Entropy) When an internal constraint on a system is removed (as with our two systems before), the total entropy may stay the same or increase. Most of the time, it increases. 19
Second Law of Thermodynamics Einstein s Desk Third Law of Thermodynamics The entropy of a system approaches a constant value as the temperature approaches zero. Real life: plots of (most) thermodynamic variables come in flat as T 0. 20
Laws of Thermodynamics 0) Thermal equilibrium means same temperature. 1) Heat is energy. (Conserve it.) 2) Entropy increases or stays same. 3) Entropy approaches constant at absolute zero Fundamental assumption: Today A closed system is equally likely to be in any accessible state. Ensemble average Two systems in thermal contact Entropy Temperature 21
Today 0 th Law: with 1 st Law: Heat is energy 2 nd Law: Left alone, entropy will not decrease 3 rd Law: Entropy approaches a constant as T 0. 22