Lecture 27: Entropy and Information Prof. WAN, Xin

Transcription

1 General Pysis I Leture 27: Entropy and Information Prof. WAN, Xin xinwan@zju.edu.n ttp://zimp.zju.edu.n/~xinwan/

2 1st & 2nd Laws of ermodynamis e 1st law speifies tat we annot get more energy out of a yli proess by work tan te amount of energy we put in. U W e 2nd law states tat we annot break even beause we must put more energy in, at te iger temperature, tan te net amount of energy we get out by work. W 1 arnot 1

3 Carnot s Engine

4 Effiieny of a Carnot Engine All Carnot engines operating between te same two temperatures ave te same effiieny.

5 An Equality Now putting in te proper signs, positive Carnot Cyle 0 d negative 0

6 A Sum of Carnot Cyles P adiabats,i Any reversible proess an be approximated by a sum of Carnot yles, ene i, i, i, i, i 0 C d 0,i

7 Clausius Definition of Entropy Entropy is a state funtion, te ange in entropy during a proess depends only on te end points and is independent of te atual pat followed. C 2 2 C ds ds 12 d C 1, 2 reversible ds,21 C ds 0 1 C 1 S 2 S 1 ds ds C1,1 2 C2,21 C 2 ds,1 2

8 Return to Inexat Differential Assume (2,1) (1,1) dg (2,2) (2,1) dx x y dx dy x y dy 1 2ln 2 (1,2) (1,1) (2,2) (1,2) dx x y dy ln 2 1 Note: df dg x dx x dy y is an exat differential. Integrating fator f ( x, y) ln x ln y f 0

9 Bak to te First Law Heat is pat dependent. d du Pd erefore, 1/ is really te integrating fator for te differential form of eat. Now we an reast te 1st law of termodynamis as du ds Pd Entropy is also a state funtion, as is te internal energy or volume.

10 Entropy of an Ideal Gas (1 mole) p(, ) R U mol ( ) C fr 2 ds 1 du pd C mol d Rd Integrating from ( 0, 0 ) to (, ) S(, ) mol S0 C ln R ln 0 0

11 Carnot s eorem No real eat engine operating between two energy reservoirs an be more effiient tan Carnot s engine operating between te same two reservoirs. positive negative e' 1 ' ' 1 ' ' Wat does tis mean? Still, for any engine in a yle (S is a state funtion!) ds 0 0

12 Counting te Heat Bats in S ' ' > 0 S gas ds 0 after a yle S ' ' < 0 S S S gas S ' 0 ' 0

13 Counting te Heat Bats in S ' ' > 0 S gas ds 0 after a yle S ' ' < 0 e total entropy of an isolated system tat undergoes a ange an never derease.

14 Example 1: Clausius Statement S S S S S 0 Irreversible!

15 Example 2: Kelvin Statement S 0 Irreversible!

16 Speifi Heat Note: Last time we defined molar speifi eat. In pysis, we also use speifi eat per partile.

17 Example 3: Mixing Water Example 3: Mixing Water A A B B A B A < B B A B B A A B A B A m m m m m m A A A m : m B B B :

18 Example 3: Mixing Water A B S A < B A B A B A B m : ln Ad S A ma 0 A A m : ln Bd SB mb 0 B B For simpliity, assume S A S B m A m mln B 2 A m B 0 / 2, A B Irreversible!

19 e Seond Law in terms of Entropy e total entropy of an isolated system tat undergoes a ange an never derease. If te proess is irreversible, ten te total entropy of an isolated system always inreases. In a reversible proess, te total entropy of an isolated system remains onstant. e ange in entropy of te Universe must be greater tan zero for an irreversible proess and equal to zero for a reversible proess. ΔS Universe 0

20 Order versus Disorder Isolated systems tend toward disorder and tat entropy is a measure of tis disorder. Ordered: all moleules on te left side Disordered: moleules on te left and rigt

21 Example 4: Free Expansion U W? 0 S 0 We an only alulate S wit a reversible proess! In tis ase, we replae te free expansion by te isotermal proess wit te same initial and final states. S i f d i f Pd i f nrd f nr ln 0 i Irreversible!

22 Entropy: A Measure of Disorder Entropy: A Measure of Disorder ln 2 ln B i f B Nk Nk S W k S B ln N m f f W N m i i W N i f i f W W We assume tat ea moleule oupies some mirosopi volume m. suggesting (Boltzmann)

23 Information and Entropy (1927) Bell Labs, Ralp Hartley Measure for information in a message Logaritm: 8 bit = 2 8 = 256 different numbers (1940) Bell Labs, Claude Sannon A matematial teory of ommuniation Probability of a partiular message But tere is no information. You are not winning te lottery.

24 Information and Entropy (1927) Bell Labs, Ralp Hartley Measure for information in a message Logaritm: 8 bit = 2 8 = 256 different numbers (1940) Bell Labs, Claude Sannon A matematial teory of ommuniation Probability of a partiular message Now tat s someting. Okay, you are going to win te lottery.

25 Information and Entropy (1927) Bell Labs, Ralp Hartley Measure for information in a message Logaritm: 8 bit = 2 8 = 256 different numbers (1940) Bell Labs, Claude Sannon A matematial teory of ommuniation Probability of a partiular message Information ~ - log (probability) ~ negative entropy S infomation i P i log P i

26 It is already in use under tat name. and besides, it will give you great edge in debates beause nobody really knows wat entropy is anyway Jon von Neumann

27 Maxwell s Demon o determine weter to let a moleule troug, te demon must aquire information about te state of te moleule. However well prepared, te demon will eventually run out of information storage spae and must begin to erase te information it as previously gatered. Erasing information is a termodynamially irreversible proess tat inreases te entropy of a system.

28 Information Redues Entropy For irreversible proesses, S > 0. Erasure is irreversible, so S Erasure > 0. So learn to remember, wi osts n bit of information, redues entropy. Example: 1 bit Knowing noting: p 1 = 1/2, p 2 = 1/2. S = log 2. Knowing wi way: p 1 = 1, p 2 = 0. S = 0. So, S = - log 2 due to te 1 bit of information.

29 Landauer s Priniple & erifiation Computation needs to involve eat dissipation only wen you do someting irreversible wit te information. Lutz group (2012) k B ln

30 For ose Wo Are Interested Reading (downloadable from my website): Carles Bennett and Rolf Landauer, e fundamental pysial limits of omputation. Antoine Bérut et al., Experimental verifiation of Landauer s priniple linking information and termodynamis, Nature (2012). Set Lloyd, Ultimate pysial limits to omputation, Nature (2000). Dare to adventure were you ave not been!