Lecture 46 Statisticalinterpretation of entropy·Microstates· Dominance of certain macrostates.Boltzmannconnection· Statistical meaning of Entropy
Lecture 46 Statistical interpretation of entropy • Microstates • Dominance of certain macrostates • Boltzmann connection • Statistical meaning of Entropy
Statistical Interpretation of Entropyand the Second Law. The microstate of a system would be specified ingiving the position and velocity of every particle (ormolecule).: The macrostate of a system is specified by givingthe macroscopic properties of the system-thetemperature, pressure, number of moles, and so on. In reality, we can know only the macrostate of asystem
Statistical Interpretation of Entropy and the Second Law • The microstate of a system would be specified in giving the position and velocity of every particle (or molecule). • The macrostate of a system is specified by giving the macroscopic properties of the system-the temperature, pressure, number of moles, and so on. • In reality, we can know only the macrostate of a system. 2
三正Flipping coins二正一反·Abasicassumptionbehindthestatistical approach is that eachmicrostate is equally probable: Some microstates are the same,we can not distinguish these一正二反states.The distribution of microstates isnot uniform.三反
Flipping coins • A basic assumption behind the statistical approach is that each microstate is equally probable. • Some microstates are the same, we can not distinguish these states. • The distribution of microstates is not uniform. 3 二 正 一 反 三 反 一 正 二 反 三 正
Probabilities of VariousMacrostates for 100 Coin TossesMacrostateNumberofHeadsTailsProbabilityMicrostates7.9×10-31010017.9×10-291.0×1029911.4×10-171.7×101390104.2×10-105.4×102020801.4×102860400.016.1 × 102855450.051.0×102950500.086.1 × 102845550.051.4 × 102840600.014.2×10-105.4×102020801.4×10-171.7×101310907.9×10-29991.0×10217.9×10-31010014
Probabilities of Various Macrostates for 100 Coin Tosses 4
When there are more coins? When there are more coins (particles), N, thenumber of microstate increase exponentially.However,the most probable macrostateis the onewith number of heads sharplypeaked at N/2 andthe number of tails sharply peaked at N/2. This is the result of statistics.: The total number of possible microstates,W = 2N is similar to N!/(N/2)!(N/2)!S
When there are more coins 5