{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "" 1 16 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 23 "Entropy and Temperature" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 227 "This w orksheet contains calculations that illustrate the statistical mechani cs definition of entropy. Two coupled Einstein solids are considered f ollowing the explanations by Thomas A Moore and Daniel V Schroeder (Am . J. Phys. " }{TEXT 257 2 "65" }{TEXT -1 13 ", 26 (1997))." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 741 "In macroscopic t hermodynamics entropy is defined as the quantity that increases by Q/T when a thermodynamic system receives the energy amount Q by heating w hile at temperature T. This is not intuitive, and the additional expla nations about entropy measuring order in the system are also ad-hoc. T his definition of entropy is referred to as the Clausius definition. B oltzmann is the originator of the statistical mechanics definition. In the Einstein solid the thermodynamic system is simplified by consider ing a given number of oscillators at fixed locations. Each atom has th ree degrees of freedom, and the oscillators can take up energy in quan tized units. The quantum zero-point energy of the oscillators is irrel evant for the discussion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 520 "The connection between the macroscopic and micros copic descriptions is as follows. Macroscopically the system is descri bed by a a few variables (For a tank of an atomic gas: Number of atoms , Volume of the gas, and total internal energy U; other variables such as pressure and temperature follow from the ideal gas law). Microscop ically there can be many states that are consistent with this macrosta te. The number of consistent microstates is called the multiplicity. D ifferent macrostates have different multiplicities." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 435 "If a system is isolate d in a given macrostate it is equally likely to be in any of the consi stent microstates - they have all equal probability [fundamental assum ption of stat mech]. The probability equals the inverse multiplicity f or that macrostate. This assumption leads to the second law of thermod ynamics, namely that macroscopic objects exhibit irreversible behaviou r. To prove this point we consider two coupled Einstein solids." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "First we \+ calculate the multiplicity for an Einstein solid (for a derivation see : H B Callen, " }{TEXT 258 54 "Thermodynamics and an Introduction to T hermostatistics" }{TEXT -1 10 ", p. 334)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "We have a system of " }{TEXT 262 1 "N" }{TEXT -1 53 " one-dimensional oscillators (the number of atoms \+ is " }{TEXT 263 1 "N" }{TEXT -1 45 "/3), each oscillator can be in a s tate where " }{TEXT 264 1 "n" }{TEXT -1 29 " units of energy are store d, " }{TEXT 259 1 "n" }{TEXT -1 50 "=0,1,2,...; the total energy of th e system equals " }{TEXT 265 1 "q" }{TEXT -1 18 " units of energy (" } {TEXT 266 1 "q" }{TEXT -1 53 " is also a non-negative integer). The un it of energy " }{TEXT 260 1 "E" }{TEXT -1 4 " = h" }{TEXT 261 2 " f" } {TEXT -1 116 " , where f is he natural frequency of the oscillator and h is Planck's constant. The multiplicity for a system with " }{TEXT 268 1 "N" }{TEXT -1 42 " degrees of freedom and a total number of " } {TEXT 267 1 "q" }{TEXT -1 85 " units of energy can be shown to be (at \+ least try to verify this by counting for low " }{TEXT 270 1 "N" } {TEXT -1 2 ", " }{TEXT 269 1 "q" }{TEXT -1 2 "):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plots):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 32 "Omega:=(N,q)->binomial(q+N-1,q);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%&OmegaGR6$%\"NG%\"qG6\"6$%)operatorG%&arrowGF) -%)binomialG6$,(9%\"\"\"9$F2!\"\"F2F1F)F)F)" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 140 "A system of 3 oscillators with 3 units of energy share d can be in a state (300), (210), (111), (012), (030), ... a total of \+ 10 combinations." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Omega(3 ,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 830 "The two coupled Einstein solids represent an interacti ng system; we label the two by A and B. We can consider a macrostate w here the total energy U is split between A and B as U = U_A + U_B. The reason for being able to specify U_A and U_B lies in the fact that th e coupling between the two solids is weak: while within the solids the systems are fluctuating between the microstates consistent with the m acrostates (e.g., U_A, N_A), the transfer of energy between A and B ha ppens on a longer timescale and it is possible to know U_A and U_B. On e calculates the multiplicities for A and B, and takes the product of \+ these to find the multiplicity for the coupled system. This is true si nce the two solids are independent of each other, and any microstate i n A can combine with any microstate in B (for given macrostates in A a nd B)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 257 "What we need now is a counting scheme: for given N_A, N_B and tot al number of excitation quanta q_t, we wish to find out how many micro states there are for a given combination where q_A, and q_B units of e nergy stored in A, B respectively (q_A + q_B = q_t)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "N_A:=300; N_B:=200;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$N_AG\"$+$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$N_BG \"$+#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "q_t:=20;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$q_tG\"#?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "We work with lists in Maple (lists are ordered sets), an d define ourselves a function to add an element to a list:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ladd:=(L,e)->[op(L),e];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%laddGR6$%\"LG%\"eG6\"6$%)operatorG%&arrow GF)7$-%#opG6#9$9%F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " ladd([1,2,3],4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"\"\"\"#\"\"$ \"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "O_A:=[]: O_B:=[]: \+ O_t:=[]: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "for q_A from 0 to q_t \+ do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "q_B:=q_t-q_A;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "o_A:=Omega(N_A,q_A);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "o_B:=Omega(N_B,q_B);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "o_t:=o_A*o_B;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "O_A:=ladd(O_ A,o_A); O_B:=ladd(O_B,o_B); O_t:=ladd(O_t,o_t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "The lists can be printed. If q is large, however, the lists are very long, and \+ we have commented out the statement." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "#O_A;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "#O_ B;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "#O_t;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "listplot(O_t);" }}{PARA 13 "" 1 "" {GLPLOT2D 631 225 225 {PLOTDATA 2 "6#-%'CURVESG6#777$$\"\"\"\"\"!$\"+R 04#3\"\"#>7$$\"\"#F*$\"+:;jkH\"#?7$$\"\"$F*$\"+I\"*p))Q\"#@7$$\"\"%F*$ \"+vL:ZK\"#A7$$\"\"&F*$\"+.(*)e$>\"#B7$$\"\"'F*$\"+&fD#f()FE7$$\"\"(F* $\"+bd)47$\"#C7$$\"\")F*$\"+`$Rt'*)FP7$$\"\"*F*$\"+:*z,6#\"#D7$$\"#5F* $\"+Lo,2TFen7$$\"#6F*$\"+-,]ZmFen7$$\"#7F*$\"+c7dj*)Fen7$$\"#8F*$\"+4( p^+\"\"#E7$$\"#9F*$\"+YKJB$*Fen7$$\"#:F*$\"+RD+$3(Fen7$$\"#;F*$\"+EZjR VFen7$$\"# " 0 "" {MPLTEXT 1 0 33 "MSD:=proc(L) local n,i,av ,dev,wt;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "n:=nops(L); wt:=add(L[i ],i=1..n):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "av:=add((i)*L[i],i=1. .n)/wt:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "dev:=evalf(sqrt(add((i-a v)^2*L[i],i=1..n)/wt));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "print(`a verage: `,av-1,` deviation: `,dev);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "MSD(O_t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&%*average:~G\"#7%-~deviation:~G$\"+vu/ KA!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 237 "We calculated the expe ctation value of the distance squared from the average value and took \+ the square root to obtain a distance. The expectation values for the a verage are normalized by dividing the sums by the sum over all data po ints." }}{PARA 0 "" 0 "" {TEXT -1 17 "N_A=300, N_B=200:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "We see from repeated calculations with increased q: " }}{PARA 0 "" 0 "" {TEXT -1 18 "q=100 : 60, dev 5.4" }}{PARA 0 "" 0 "" {TEXT -1 19 "q=200: 120, dev 8.2" }} {PARA 0 "" 0 "" {TEXT -1 20 "q=300: 180, dev 10.7" }}{PARA 0 "" 0 "" {TEXT -1 20 "q=400: 240, dev 13.1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 20 "q=800: 480, dev 22.3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 294 "For fixed number of degr ees of freedom we have increased the amount of energy deposited. The c ombination of energy quanta distributed between A and B follows some o rder: q_A = 6/10 * q_t is the location with the largest multiplicity f or the choice of N_A vs N_B. How does the width grow with q?" }}{PARA 0 "" 0 "" {TEXT -1 74 "This has to be considered carefully. The number of states increases like q" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 " For 20/30 we get " }}{PARA 0 "" 0 "" {TEXT -1 20 "q=800: 480. dev 56.6" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "v1:=56.6/480;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v1G$\"+nm;z6!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "v2:=111.5/960;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# v2G$\"+L$e9;\"!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "v2/v1; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+;L#)\\)*!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "evalf(1/sqrt(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+5y1rq!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "v3:=166.3/1440;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v3G$\"+66'[: \"!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "v3/v1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+W^(Qz*!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "evalf(1/sqrt(3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"+$p-Nx&!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "v4:=13.1/2 40;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v4G$\"+LLLea!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "v4/v1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+j_(*GY!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "eval f(1/sqrt(4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++]!#5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "v8:=22.3/480;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#v8G$\"+LL$ek%!#6" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 6 "v8/v1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+F$H*RR !#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "evalf(1/sqrt(8));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+0R`NN!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 162 "The relative width is not decreasing as 1/sqrt(q), bu t we are also not in the limit of large q, yet. If N was larger than q , the rel. width would go as 1/sqrt(N)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 694 "For many degrees of freedom (N appr oaching 10^24, and implying also a large number of energy units deposi ted) the distribution becomes very narrowly peaked. Thus, there is an \+ almost uniquely defined macrostate with a very high multiplicity compa red to the others. The second law of thermodynamics (that states that \+ entropy increases) can be cast into the form that the system evolves a lways towards the macrostate with the largest multiplicity. It happens since in the course of fluctuations the system is bound to evolve int o the state with the largest probability. Given that all microstates a re assumed to be equally probable, it must evolve towards the macrosta te with largest multiplicity." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 305 "Entropy is defined now as the log arithm of the multiplicity. This ensures the additivity of entropies i n weakly coupled systems (since their combined multiplicity was given \+ as a product of the individual ones). In addition, the entropy definit ion contains Boltzmann's constant as a proportionality factor." }} {PARA 0 "" 0 "" {TEXT -1 15 "S = k ln(Omega)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "For plotting purposes we chose k=1 (a unit system). We can calculate the entropies for the subsystem s and the total." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "P1:=lis tplot(map(log,O_A),color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "P2:=listplot(map(log,O_B),color=blue):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 39 "P3:=listplot(map(log,O_t),color=green):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "display(P1,P2,P3);" }}{PARA 13 "" 1 "" {GLPLOT2D 684 262 262 {PLOTDATA 2 "6%-%'CURVESG6$777$$\"\" \"\"\"!F*7$$\"\"#F*$\"+vCy.d!\"*7$$\"\"$F*$\"+cXxr5!\")7$$\"\"%F*$\"+H g&H`\"F67$$\"\"&F*$\"+t)*pl>F67$$\"\"'F*$\"+_)ekP#F67$$\"\"(F*$\"+$39$ pFF67$$\"\")F*$\"+y:3ZJF67$$\"\"*F*$\"+*>A=^$F67$$\"#5F*$\"+>(4^'QF67$ $\"#6F*$\"+Q`=3UF67$$\"#7F*$\"+SI0UXF67$$\"#8F*$\"+m;an[F67$$\"#9F*$\" +\\qM&=&F67$$\"#:F*$\"+N;1'\\&F67$$\"#;F*$\"+9f>+eF67$$\"#)4 'F67$$\"#=F*$\"+$>Z/R'F67$$\"#>F*$\"+%>+tn'F67$$\"#?F*$\"+N91fpF67$$\" #@F*$\"+)yrfF67$F8$\"+Kc$Gs&F67$F=$\"+E\"H'oaF67$FB$ \"+&>C)3_F67$FG$\"+9;.V\\F67$FL$\"+I\"33n%F67$FQ$\"+RWk\"R%F67$FV$\"+! H\\\\5%F67$Fen$\"+k!G+\"QF67$Fjn$\"+\\J01NF67$F_o$\"+)z@?>$F67$Fdo$\"+ tSpmGF67$Fio$\"+ru\\GDF67$F^p$\"+?CPv@F67$Fcp$\"+7U]/=F67$Fhp$\"+]I\"= T\"F67$F]q$\"+%4v%3**F07$Fbq$\"+ntJ)H&F07$FgqF*-F\\r6&F^rF*F*F_r-F$6$7 7Fdr7$F,$\"+[?<'y'F67$F2$\"+[FcVqF67$F8$\"+h;zbsF67$F=$\"+***GVV(F67$F B$\"+ZIG&e(F67$FG$\"+(pXBr(F67$FL$\"+3(*)y\"yF67$FQ$\"+PmY.zF67$FV$\"+ 5!f+(zF67$Fen$\"+-M@=!)F67$Fjn$\"+*=1\"[!)F67$F_o$\"+lMcf!)F67$Fdo$\"+ B6/_!)F67$Fio$\"+2\"fX-)F67$F^p$\"+M$ob(zF67$Fcp$\"+=&)p-zF67$Fhp$\"+V -E-yF67$F]q$\"+/x9owF67$Fbq$\"+rJ*))[(F6Ffq-F\\r6&F^rF*F_rF*" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "The graph shows that at equilibrium (which happens at q_ A=60) the total entropy is stationary and has a maximum." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 208 "We understand now that the evolution of the macrosystem can be described as consistent \+ with the 2nd law of thermodynamics in the form that the isolated syste m evolves towards a state with the largest entropy." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "Note that the total ent ropy S_t is fairly flat in the vicinity of the maximum as a function o f q_A." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 271 30 "How can we define temperature?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 206 "One can adopt a definition bas ed on energy flow and thermal equilibrium: two objects in thermal cont act are at the same temperature if they are in thermal equilibrium, in which case there is no spontaneous " }{TEXT 272 3 "net" }{TEXT -1 532 " flow between them. If there is a spontaneous net flow from A to \+ B, then A loses energy and is at a higher temperature, and B gains ene rgy and is at a lower temperature. Now to relate temperature to entrop y we have to find in the above figure at the equilibrium point two qua ntitites that become equal. These are the slopes of the curves for S_A (q_A) and S_B(q_A), which are equal in magnitude and opposite in sign. S_B(q_B) has the same slope as S_A(q_A), i.e., the sign is also equal . Thus, temperature has to be related to dS/dq." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 163 "On dimensional grounds ( given the choice of constant k in the definition of S) it is the inver se temperature that equals the rate of change of entropy with energy. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 824 "The \+ above figure is used by Moore and Schroeder to justify this. Assume th at you are on the right of the equilibrium point q_A=60 units, i.e., s olid A has more energy than what it would have at equilibrium. In this region the curve for S_A (red) has a positive slope that is levelling off. The (blue) curve has a negative slope that is increasing in magn itude. What does this mean? If a small amount of energy (e.g. one unit dq=1) were to pass from A to B in this region (a movement to the left in the graph, or a move towards equilibrium), the increase in S_B wou ld be bigger than the decrease in S_A. The total entropy would increas e (green curve). The second law of TD tells us that this process will \+ happen spontaneously (there are many more microstates in this region, \+ i.e., a fluctuation will find this regime easily)." }}{PARA 0 "" 0 "" {TEXT -1 253 "The steeper the slope in the entropy versus energy curve the more the system wants to obey the second law and increase its ene rgy (B), while the a shallower entropy-energy curve for the other syst em (A) means that it doesn't mind to give up some energy." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 374 "To summarize: to \+ the right of the equilibrium point A has more energy than at equilibri um and energy flows from A to B to reach equilibrium. A thus has a hig her temperature than B there. Its entropy-energy curve has a smaller s lope, while the slope for |S_B(q_A)| is larger. Thus, the inverse of \+ the temperature is proportional to the rate of change of entropy with \+ energy:" }}{PARA 0 "" 0 "" {TEXT -1 11 "1/T = dS/dU" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 171 "When the volume of the system is allowed to change (as is true for gases) one has to take pa rtial derivatives, and one cannot simply rearrange the above eq. into \+ dS=dU/T !" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 256 "For the Einstein solid we can use the above equation to calculate the temperature as T = dU/dS by using neighbouring values in the entr opy as the energy is changed by one unit up and down. A central differ ence formula is used, the spacing in U equals dq=2." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "We take a single solid \+ with 50 degrees of freedom and vary q between zero and 100" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "N:=50;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"#]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "O1:=[]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "for q from 0 to 100 do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "o1:=Omega(N,q);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "O1:=ladd(O1,o1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "S1:=map(evalf,map(log,O1)):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Now we can calculate the temperature:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "T1:=[]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for q from 2 to 99 do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "t1:=2/(S1[q+1]-S1[q-1]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "T1:=ladd(T1,t1); od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "li stplot(T1);" }}{PARA 13 "" 1 "" {GLPLOT2D 695 227 227 {PLOTDATA 2 "6#- %'CURVESG6#7^q7$$\"\"\"\"\"!$\"+-%Gpz#!#57$$\"\"#F*$\"+QgO$G$F-7$$\"\" $F*$\"+3-vyOF-7$$\"\"%F*$\"+o#z$HSF-7$$\"\"&F*$\"+/OW_VF-7$$\"\"'F*$\" +q3ccYF-7$$\"\"(F*$\"+O*en%\\F-7$$\"\")F*$\"+y-EE_F-7$$\"\"*F*$\"+_!os \\&F-7$$\"#5F*$\"+q:OhdF-7$$\"#6F*$\"+][r>gF-7$$\"#7F*$\"+]hAtiF-7$$\" #8F*$\"+O*)fAlF-7$$\"#9F*$\"+7]RonF-7$$\"#:F*$\"+w/26qF-7$$\"#;F*$\"+q 2+^sF-7$$\"#F*$\"+/E@dzF- 7$$\"#?F*$\"+#[W))=)F-7$$\"#@F*$\"+E&*))=%)F-7$$\"#AF*$\"+A>\\Z')F-7$$ \"#BF*$\"+9zxu))F-7$$\"#CF*$\"+![e35*F-7$$\"#DF*$\"+3<$eK*F-7$$\"#EF*$ \"+5Zy\\&*F-7$$\"#FF*$\"+k_zs(*F-7$$\"#GF*$\"+=I$\\***F-7$$\"#HF*$\"+^ gi@5!\"*7$$\"#IF*$\"+AMoV5Fdt7$$\"#JF*$\"+=0nl5Fdt7$$\"#KF*$\"+x>f(3\" Fdt7$$\"#LF*$\"+1?X46Fdt7$$\"#MF*$\"+nWDJ6Fdt7$$\"#NF*$\"+wG+`6Fdt7$$ \"#OF*$\"+Y/qu6Fdt7$$\"#PF*$\"+q,N'>\"Fdt7$$\"#QF*$\"+eZ&z@\"Fdt7$$\"# RF*$\"+=n^R7Fdt7$$\"#SF*$\"+0%Q5E\"Fdt7$$\"#TF*$\"+f>_#G\"Fdt7$$\"#UF* $\"+x$pRI\"Fdt7$$\"#VF*$\"+BDQD8Fdt7$$\"#WF*$\"+?JwY8Fdt7$$\"#XF*$\"+w F6o8Fdt7$$\"#YF*$\"+(*HV*Q\"Fdt7$$\"#ZF*$\"+\">D2T\"Fdt7$$\"#[F*$\"+k1 *>V\"Fdt7$$\"#\\F*$\"+c1B`9Fdt7$$\"#]F*$\"+3jWu9Fdt7$$\"#^F*$\"+*pQc\\ \"Fdt7$$\"#_F*$\"+f)3o^\"Fdt7$$\"#`F*$\"+^x&z`\"Fdt7$$\"#aF*$\"+fi3f:F dt7$$\"#bF*$\"+\\_>!e\"Fdt7$$\"#cF*$\"+QbG,;Fdt7$$\"#dF*$\"+VyNA;Fdt7$ $\"#eF*$\"+3HTV;Fdt7$$\"#fF*$\"+@9Xk;Fdt7$$\"#gF*$\"++SZ&o\"Fdt7$$\"#h F*$\"+k7[1 oTpFdt7$$\"#sF*$\"+iqpO>Fdt7$$\"#tF*$\"+\\%ev&>Fdt7$$ \"#uF*$\"+k,Ty>Fdt7$$\"#vF*$\"+zDD**>Fdt7$$\"#wF*$\"+ef3??Fdt7$$\"#xF* $\"+i0\"4/#Fdt7$$\"#yF*$\"+;nsh?Fdt7$$\"#zF*$\"+QY`#3#Fdt7$$\"#!)F*$\" +yXL.@Fdt7$$\"#\")F*$\"+7o7C@Fdt7$$\"##)F*$\"+K:\"\\9#Fdt7$$\"#$)F*$\" +e*)ol@Fdt7$$\"#%)F*$\"+_$fk=#Fdt7$$\"#&)F*$\"+oGA2AFdt7$$\"#')F*$\"+% pzzA#Fdt7$$\"#()F*$\"+_+t[AFdt7$$\"#))F*$\"+OTZpAFdt7$$\"#*)F*$\"+_?@! 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One normalizes it usual ly by the number of oscillators (to take out the size of the system). " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "C1:=[]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for q from 2 to 97 do:" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 26 "c1:=2/(T1[q+1]-T1[q-1])/N;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "C1:=ladd(C1,c1); od:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "The simple listplot graphs the specific heat capacity (a t fixed volume) as a function of energy (q)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "listplot(C1);" }}{PARA 13 "" 1 "" {GLPLOT2D 711 243 243 {PLOTDATA 2 "6#-%'CURVESG6#7\\q7$$\"\"\"\"\"!$\"+KQ1OX!#57$$\" \"#F*$\"+/_$=O&F-7$$\"\"$F*$\"+W!>u$fF-7$$\"\"%F*$\"+/\"QxP'F-7$$\"\"& F*$\"+/QVInF-7$$\"\"'F*$\"+WlC@qF-7$$\"\"(F*$\"+)Q,gE(F-7$$\"\")F*$\"+ C-AvuF-7$$\"\"*F*$\"+s>GcwF-7$$\"#5F*$\"+#*oc9yF-7$$\"#6F*$\"+s#>T&zF- 7$$\"#7F*$\"+sB0y!)F-7$$\"#8F*$\"+3(3))=)F-7$$\"#9F*$\"+S(Q$)G)F-7$$\" #:F*$\"+S3Ay$)F-7$$\"#;F*$\"+7%[(f%)F-7$$\"#F*$\"+ou1k')F-7$$\"#?F*$\"+;eH@()F-7$$\"#@F*$\"+'RxS x)F-7$$\"#AF*$\"+%y#)G#))F-7$$\"#BF*$\"+O=7o))F-7$$\"#CF*$\"+#*)\\,\"* )F-7$$\"#DF*$\"+'\\v#\\*)F-7$$\"#EF*$\"++$pd)*)F-7$$\"#FF*$\"+_D()>!*F -7$$\"#GF*$\"+cqz^!*F-7$$\"#HF*$\"+o4t\"3*F-7$$\"#IF*$\"+O?%)4\"*F-7$$ \"#JF*$\"+G:GO\"*F-7$$\"#KF*$\"++;=h\"*F-7$$\"#LF*$\"+C:m%=*F-7$$\"#MF *$\"+?f$o?*F-7$$\"#NF*$\"+?jzF#*F-7$$\"#OF*$\"+GHjZ#*F-7$$\"#PF*$\"+SZ Vm#*F-7$$\"#QF*$\"+'4lUG*F-7$$\"#RF*$\"+;>>,$*F-7$$\"#SF*$\"+OEG<$*F-7 $$\"#TF*$\"+W$)eK$*F-7$$\"#UF*$\"+o3;Z$*F-7$$\"#VF*$\"+%=\\5O*F-7$$\"# WF*$\"+cYHu$*F-7$$\"#XF*$\"+3d$pQ*F-7$$\"#YF*$\"+k9,*R*F-7$$\"#ZF*$\"+ ?Yb5%*F-7$$\"#[F*$\"+_xf@%*F-7$$\"#\\F*$\"+WI8Y*F-7$$\"#`F*$\"+?zDq%*F-7$$\" #aF*$\"+O\\$)y%*F-7$$\"#bF*$\"+;B3([*F-7$$\"#cF*$\"+gZ,&\\*F-7$$\"#dF* $\"+'>PE]*F-7$$\"#eF*$\"+%ew*4&*F-7$$\"#fF*$\"+3s/<&*F-7$$\"#gF*$\"+?W &Q_*F-7$$\"#hF*$\"+sjTI&*F-7$$\"#iF*$\"+#R[n`*F-7$$\"#jF*$\"+!y\\Ga*F- 7$$\"#kF*$\"+%)Gu[&*F-7$$\"#lF*$\"+?*RWb*F-7$$\"#mF*$\"+o7$*f&*F-7$$\" #nF*$\"+C-Cl&*F-7$$\"#oF*$\"+crPq&*F-7$$\"#pF*$\"+OsMv&*F-7$$\"#qF*$\" +%ec,e*F-7$$\"#rF*$\"+![/[e*F-7$$\"#sF*$\"+3KJ*e*F-7$$\"#tF*$\"+;mn$f* F-7$$\"#uF*$\"+Cv*yf*F-7$$\"#vF*$\"+K6+-'*F-7$$\"#wF*$\"+sI(fg*F-7$$\" #xF*$\"+KH#)4'*F-7$$\"#yF*$\"+Wxc8'*F-7$$\"#zF*$\"+[Q><'*F-7$$\"#!)F*$ \"++Wr?'*F-7$$\"#\")F*$\"+C@9C'*F-7$$\"##)F*$\"++1YF'*F-7$$\"#$)F*$\"+ /\\oI'*F-7$$\"#%)F*$\"+;?$Qj*F-7$$\"#&)F*$\"+#*f)oj*F-7$$\"#')F*$\"+g_ %)R'*F-7$$\"#()F*$\"+3mtU'*F-7$$\"#))F*$\"+!o\\bk*F-7$$\"#*)F*$\"+)Gy# ['*F-7$$\"#!*F*$\"+w8%4l*F-7$$\"#\"*F*$\"+;&HNl*F-7$$\"##*F*$\"+Sq0c'* F-7$$\"#$*F*$\"+go^e'*F-7$$\"#%*F*$\"+gY!4m*F-7$$\"#&*F*$\"+cTCj'*F-7$ $\"#'*F*$\"+)eBbm*F-" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "To plot C(T) we do :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot([[T1[j],C1[j+1]] \+ $j=1..95],view=[0..3,0..1]);" }}{PARA 13 "" 1 "" {GLPLOT2D 706 283 283 {PLOTDATA 2 "6%-%'CURVESG6$7[q7$$\"1+++-%Gpz#!#;$\"1*****R?N=O&F*7 $$\"1+++QgO$G$F*$\"1+++W!>u$fF*7$$\"1+++3-vyOF*$\"1+++/\"QxP'F*7$$\"1+ ++o#z$HSF*$\"1+++/QVInF*7$$\"1+++/OW_VF*$\"1+++WlC@qF*7$$\"1+++q3ccYF* $\"1+++)Q,gE(F*7$$\"1+++O*en%\\F*$\"1+++C-AvuF*7$$\"1+++y-EE_F*$\"1+++ s>GcwF*7$$\"1*****>0os\\&F*$\"1+++#*oc9yF*7$$\"1+++q:OhdF*$\"1+++s#>T& zF*7$$\"1+++][r>gF*$\"1+++sB0y!)F*7$$\"1+++]hAtiF*$\"1+++3(3))=)F*7$$ \"1+++O*)fAlF*$\"1+++S(Q$)G)F*7$$\"1+++7]RonF*$\"1+++S3Ay$)F*7$$\"1+++ w/26qF*$\"1+++7%[(f%)F*7$$\"1+++q2+^sF*$\"1+++cC*R`)F*7$$\"1+++3))\\)[ (F*$\"1+++Sp%=g)F*7$$\"1******p!GQs(F*$\"1+++ou1k')F*7$$\"1+++/E@dzF*$ \"1+++;eH@()F*7$$\"1*****>[W))=)F*$\"1+++'RxSx)F*7$$\"1+++E&*))=%)F*$ \"1+++%y#)G#))F*7$$\"1+++A>\\Z')F*$\"1+++O=7o))F*7$$\"1+++9zxu))F*$\"1 +++#*)\\,\"*)F*7$$\"1+++![e35*F*$\"1+++'\\v#\\*)F*7$$\"1+++3<$eK*F*$\" 1,+++$pd)*)F*7$$\"1+++5Zy\\&*F*$\"1+++_D()>!*F*7$$\"1+++k_zs(*F*$\"1,+ +cqz^!*F*7$$\"1+++=I$\\***F*$\"1+++o4t\"3*F*7$$\"1+++^gi@5!#:$\"1+++O? 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We need many oscillator s." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "N:=1000000; q_m:=200;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"(+++\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$q_ mG\"$+#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "O1:=[]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "for q from 0 to q_m do:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "o1:=Omega(N,q);" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "O1:=ladd(O1,o1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "S1:=map(evalf,map (log,O1)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Now we can calculat e the temperature:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "T1:=[] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "for q from 2 to q_m-1 \+ do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "t1:=2/(S1[q+1]-S1[q-1]);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "T1:=ladd(T1,t1); od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "listplot(T1);" }}{PARA 13 "" 1 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for a system at a given temperature it provides the pr obability to find the system in a microstate of energy E. The system i s at temperature T while in thermal contact with a heat bath (a large \+ system). This relative probability is given by exp(-E/kT), and is deri ved in texts on statistical mechanics." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 302 "The results from the previous sectio n can be used to demonstrate this behaviour. We can consider a coupled system of a N=1000 oscillator Einstein solid coupled to a single osci llator. We vary the temperature by changing the amount of energy units shared between the single oscillator and the heat bath." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plots):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Omega:=(N,q)->binomial(q+N-1,q); ladd:=(L,e)- >[op(L),e];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&OmegaGR6$%\"NG%\"qG6 \"6$%)operatorG%&arrowGF)-%)binomialG6$,(9%\"\"\"9$F2!\"\"F2F1F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%laddGR6$%\"LG%\"eG6\"6$%)operator G%&arrowGF)7$-%#opG6#9$9%F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "The 'solid' A is the single oscillator, while B represents the hea t bath." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "N_A:=1; N_B:=100 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$N_AG\"\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$N_BG\"%+5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "q_t:=100;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$q_tG\"$+\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "O_A:=[]: O_B:=[]: O_t:=[]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "for q_A from 0 to q_t do: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "q_B:=q_t-q_A;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "o_A:=Omega(N_A,q_A);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "o_B:=Omega(N_B,q_B);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "o_t:=o_A*o_B;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "O_A:=ladd(O_ A,o_A); O_B:=ladd(O_B,o_B); O_t:=ladd(O_t,o_t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "O_t: =map(evalf@log,O_t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "P1: =listplot(O_t,view=[0..30,250..340],color=blue): display(P1);" }} {PARA 13 "" 1 "" {GLPLOT2D 627 297 297 {PLOTDATA 2 "6$-%'CURVESG6$7aq7 $$\"\"\"\"\"!$\"+voH=L!\"(7$$\"\"#F*$\"+$F-7$$ \"\"(F*$\"+ph3tJF-7$$\"\")F*$\"+-Bb[JF-7$$\"\"*F*$\"+N1#R7$F-7$$\"#5F* $\"+@+>*4$F-7$$\"#6F*$\"+(GfV2$F-7$$\"#7F*$\"+KsU\\IF-7$$\"#8F*$\"+KER CIF-7$$\"#9F*$\"+JUD**HF-7$$\"#:F*$\"+Z2,uHF-7$$\"#;F*$\"+n3m[HF-7$$\" #F*$\"+b#f>(GF-7$$\"#?F*$\"+ H+ G/BF-7$$\"#TF*$\"+o/tvAF-7$$\"#UF*$\"+5J-ZAF-7$$\"#VF*$\"+Gr:=AF-7$$\" #WF*$\"+e'H\"*=#F-7$$\"#XF*$\"+Ox$*f@F-7$$\"#YF*$\"+$Hy08#F-7$$\"#ZF*$ \"+Z\"[55#F-7$$\"#[F*$\"+.SMr?F-7$$\"#\\F*$\"+QCYT?F-7$$\"#]F*$\"+,**R 6?F-7$$\"#^F*$\"+.F:\")>F-7$$\"#_F*$\"+2qr]>F-7$$\"#`F*$\"+A))3?>F-7$$ \"#aF*$\"+!*RE*)=F-7$$\"#bF*$\"+z\"Q#e=F-7$$\"#cF*$\"+ro+F=F-7$$\"#dF* $\"+\\`c&z\"F-7$$\"#eF*$\"+!o3Rw\"F-7$$\"#fF*$\"+4<.K!f7F-7$$\"#uF*$\"+Qz*HA\"F-7$$\"#vF*$\"+x?i'=\"F-7$$\"#wF*$\"+j&y)\\6 F-7$$\"#xF*$\"+.Ev76F-7$$\"#yF*$\"+%>G_2\"F-7$$\"#zF*$\"+ozGP5F-7$$\"# !)F*$\"+[,8*)**!\")7$$\"#\")F*$\"+,m#3g*Fcdl7$$\"##)F*$\"+D@u2#*Fcdl7$ $\"#$)F*$\"+/li4))Fcdl7$$\"#%)F*$\"+SC?1%)Fcdl7$$\"#&)F*$\"+64;(*zFcdl 7$$\"#')F*$\"+%Rb@e(Fcdl7$$\"#()F*$\"+'f%zgrFcdl7$$\"#))F*$\"+zJjKnFcd l7$$\"#*)F*$\"+H(frH'Fcdl7$$\"#!*F*$\"+s3y`eFcdl7$$\"#\"*F*$\"+R)*z,aF cdl7$$\"##*F*$\"+YoQS\\Fcdl7$$\"#$*F*$\"+fp`oWFcdl7$$\"#%*F*$\"+C!3])R Fcdl7$$\"#&*F*$\"+/`A)[$Fcdl7$$\"#'*F*$\"+pp7wHFcdl7$$\"#(*F*$\"+Ig*eW #Fcdl7$$\"#)*F*$\"+(Q]M*=Fcdl7$$\"#**F*$\"+)GOBJ\"Fcdl7$$\"$+\"F*$\"+z _v2p!\"*7$$\"$,\"F*F*-%'COLOURG6&%$RGBGF*F*$\"*++++\"Fcdl-%%VIEWG6$;F* Fet;$\"$]#F*$\"$S$F*" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 491 "The plot is shift ed by one unit on the horizontal axis, i.e., the point at (1,335) corr esponds to q_A=0. The graph shows the logarithm of the probability for the microstate (its multiplicity) as a function of the number of unit s of energy. The function falls off exponentially which implies a line ar fall-off of the logarithm. Thus, we have demonstrated the Boltzmann factor variation with E. [At the high end of the curve near q_A=q_t o ne can see a deviation due to the finite number q_t]. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 307 "Next we observe the change with temperature. We estimate the temparature of the heat bath at the centre of the graph shown above (it varies a little due to the finiteness of the calculation, i.e., the removal of 1-30 units of ene rgy out of a bath of 1000 oscillators that share 100 units is not negl igible)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Temp1:=1/(O_t[1 4]-O_t[15]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&Temp1G$\"+@$=9'R!#5 " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Now we quadruple the number o f energy units:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "q_t:=400; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$q_tG\"$+%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "O_A:=[]: O_B:=[]: O_t:=[]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "for q_A from 0 to q_t do:" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 13 "q_B:=q_t-q_A;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "o_A:=Omega(N_A,q_A);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "o_B:= Omega(N_B,q_B);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "o_t:=o_A*o_B;" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "O_A:=ladd(O_A,o_A); O_B:=ladd(O_B, o_B); O_t:=ladd(O_t,o_t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "O_t:=map(evalf@log,O_t):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "P2:=listplot(O_t,view=[0.. 30,790..840],color=green): display(P2);" }}{PARA 13 "" 1 "" {GLPLOT2D 627 297 297 {PLOTDATA 2 "6$-%'CURVESG6$7]dl7$$\"\"\"\"\"!$\"+]G%\\L)! \"(7$$\"\"#F*$\"+lBUA$)F-7$$\"\"$F*$\"++S))4$)F-7$$\"\"%F*$\"+(pFtH)F- 7$$\"\"&F*$\"+(R`ZG)F-7$$\"\"'F*$\"+U5;s#)F-7$$\"\"(F*$\"+u0bf#)F-7$$ \"\")F*$\"+L>#pC)F-7$$\"\"*F*$\"+h]FM#)F-7$$\"#5F*$\"+(*)4;A)F-7$$\"#6 F*$\"+#QE*3#)F-7$$\"#7F*$\"+cWA'>)F-7$$\"#8F*$\"+eS]$=)F-7$$\"#9F*$\"+ G^wq\")F-7$$\"#:F*$\"+.w+e\")F-7$$\"#;F*$\"+A9BX\")F-7$$\"#\")F-7$$\"#>F*$\"+E.z1\")F-7$$\"#?F*$\"+,*QR4)F- 7$$\"#@F*$\"+2&o53)F-7$$\"#AF*$\"+\"3z\"o!)F-7$$\"#BF*$\"+e0Fb!)F-7$$ \"#CF*$\"+vGMU!)F-7$$\"#DF*$\"+nfRH!)F-7$$\"#EF*$\"+o(Hk,)F-7$$\"#FF*$ \"+9UW.!)F-7$$\"#GF*$\"+Q#R/*zF-7$$\"#HF*$\"+uZTxzF-7$$\"#IF*$\"+b2Pkz F-7$$\"#JF*$\"+;rI^zF-7$$\"#KF*$\"+)yB#QzF-7$$\"#LF*$\"+.27DzF-7$$\"#M F*$\"+%z(*>\"zF-7$$\"#NF*$\"+#*\\&))*yF-7$$\"#OF*$\"+GAp&)yF-7$$\"#PF* $\"+M%4D(yF-7$$\"#QF*$\"+QlIfyF-7$$\"#RF*$\"+rM3YyF-7$$\"#SF*$\"+k,%G$ yF-7$$\"#TF*$\"+Vld>yF-7$$\"#UF*$\"+RDH1yF-7$$\"#VF*$\"+!3))Hz(F-7$$\" #WF*$\"+%4j'zxF-7$$\"#XF*$\"+2vJmxF-7$$\"#YF*$\"+Z7&Hv(F-7$$\"#ZF*$\"+ SUcRxF-7$$\"#[F*$\"+8k:ExF-7$$\"#\\F*$\"+\"pFFr(F-7$$\"#]F*$\"+**zF*p( F-7$$\"#^F*$\"+js!eo(F-7$$\"#_F*$\"+1aJswF-7$$\"#`F*$\"+_B!)ewF-7$$\"# aF*$\"+D!o_k(F-7$$\"#bF*$\"+[BrJwF-7$$\"#cF*$\"+V_8=wF-7$$\"#dF*$\"+Km `/wF-7$$\"#eF*$\"+Ok\"4f(F-7$$\"#fF*$\"+xXFxvF-7$$\"#gF*$\"+w4hjvF-7$$ \"#hF*$\"+_b#*\\vF-7$$\"#iF*$\"+D#=i`(F-7$$\"#jF*$\"+8*)[AvF-7$$\"#kF* $\"+Ovt3vF-7$$\"#lF*$\"+7S'\\\\(F-7$$\"#mF*$\"+e#o6[(F-7$$\"#nF*$\"+\" >]tY(F-7$$\"#oF*$\"+G(4NX(F-7$$\"#pF*$\"+%yY'RuF-7$$\"#qF*$\"+w7wDuF-7 $$\"#rF*$\"+M(F-7$$\"#xF*$\"+% oBzK(F-7$$\"#yF*$\"+[X&QJ(F-7$$\"#zF*$\"+h?w*H(F-7$$\"#!)F*$\"+Lhk&G(F -7$$\"#\")F*$\"+um]rsF-7$$\"##)F*$\"+$fVtD(F-7$$\"#$)F*$\"+'zcJC(F-7$$ \"#%)F*$\"+$>Y*GsF-7$$\"#&)F*$\"+!p6Z@(F-7$$\"#')F*$\"+$>`/?(F-7$$\"#( )F*$\"+21<'=(F-7$$\"#))F*$\"+QQ'=<(F-7$$\"#*)F*$\"+!zKv:(F-7$$\"#!*F*$ \"+nt/(F-7$$\"#)*F*$\"+5ZWFqF-7$$\"#**F*$\"+pa'G,(F-7$$ \"$+\"F*$\"+b3E)*pF-7$$\"$,\"F*$\"+i2j$)pF-7$$\"$-\"F*$\"+(3v*opF-7$$ \"$.\"F*$\"+BPHapF-7$$\"$/\"F*$\"+mleRpF-7$$\"$0\"F*$\"+4N&[#pF-7$$\"$ 1\"F*$\"+WW45pF-7$$\"$2\"F*$\"+j#4`*oF-7$$\"$3\"F*$\"+cy\\!)oF-7$$\"$4 \"F*$\"+;,mloF-7$$\"$5\"F*$\"+Ifz]oF-7$$\"$6\"F*$\"+(=0f$oF-7$$\"$7\"F *$\"+xx)4#oF-7$$\"$8\"F*$\"+%eVg!oF-7$$\"$9\"F*$\"+(\\s5z'F-7$$\"$:\"F *$\"++W2wnF-7$$\"$;\"F*$\"+y\"\\5w'F-7$$\"$<\"F*$\"+:n*fu'F-7$$\"$=\"F *$\"+%*o\"4t'F-7$$\"$>\"F*$\"+(f4er'F-7$$\"$?\"F*$\"+/Zn+nF-7$$\"$@\"F *$\"+)47bo'F-7$$\"$A\"F*$\"+c;KqmF-7$$\"$B\"F*$\"+dK5bmF-7$$\"$C\"F*$ \"+!yc)RmF-7$$\"$D\"F*$\"++@eCmF-7$$\"$E\"F*$\"+%4z#4mF-7$$\"$F\"F*$\" +Pw%Rf'F-7$$\"$G\"F*$\"+-weylF-7$$\"$H\"F*$\"+i))>jlF-7$$\"$I\"F*$\"+! 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units vertical vs 30 horizontal" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 279 "We quadrupled the number of energ y units in the heat bath made up of 1000 oscillators and the slope in \+ the log(Omega) plot halved. If Omega is proportional to exp(-E/kT) as \+ claimed we need to show that the temperature increases like the square root of the number of energy units." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Temp2:=1/(O_t[14]-O_t[15]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&Temp2G$\"+'*4^Qy!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 275 "The temperature of the heat bath is approximately twice \+ of what it was in the first case. Thus, we have demonstrated that the \+ probability distribution as calculated from the multiplicity associate d with the heat bath follows the dependence given by the Boltzmann fac tor exp(-" }{TEXT 275 1 "E" }{TEXT -1 2 "/k" }{TEXT 276 1 "T" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 273 43 "Thermal behaviour of a two-state paramagnet" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 539 "Moore and Schroed er discuss this problem as a an example that leads to further implicat ions of the theoretical definition of temperature given above. In conv entional thermodynamics temperature is the measure of the amount of ki netic motion (linear motion of atoms in a gas, vibrational motion of d iatomic or other molecules, vibrations of stationary atoms in the Eins tein solid, etc.). In the case of a magnet the degrees of freedom are \+ contained in the spin orientation of individual atoms with respect to \+ an externally defined direction." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 484 "The energy of an atom with an outer elec tron with unpaired spin in an outer magnetic field depends on the orie ntation of the electron spin, which gives rise to a magnetic moment (i f the electron is in an s state,or in a state with zero magnetic quant um number there is no contribution from orbital angular momentum to th e magnetic moment). The magnetic moment vector is opposite to the spin vector due to the negative charge of the electron (see e.g., Wolfson \+ and Pasachoff, p. 1098)." }}{PARA 0 "" 0 "" {TEXT -1 962 "The energy o f a magnetic dipole in an external field is given as U = - mu . B , an d therefore the state with a magnetic dipole oriented in the same dire ction as the external field has a lower energy than the state that has a counteraligned magnetic dipole moment. The magnetic dipole moment i s defined such that the torque produced by the magnetic field on it ac ts to align the dipole with the magnetic field vector. (The evidence f or this is obtained from tracing the magnetic field of a big solenoid \+ or bar magnet by placing a compass at various locations: near the Nort h and South poles it is evident how the compass needle (its dipole) al igns with the field; at the sides of the 'big' magnet it apparently co unteraligns: this is a result of the inhomogeneity of the field: the f ield lines outside the bar magnet run in the opposite direction to for m closed loops - this causes two permanent magnets side-by-side to sna p together in a counteraligned fashion)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 309 "For our problem this complication does not exist. The external field is homogeneous. The total potentia l energy of a system of N dipoles depends on how many are aligned (poi nting up) N_u and how many are counteraligned (pointing down) N_d, wit h N_u + N_d = N . If we chose units such that mu=1 and B=1 we have" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "The energ ies of the individual up and down states for the dipoles:" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "E_u = -mu B, E_d = mu B." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "U = N_u E_u + N_d E_d = mu B (N_d - N_u) = mu B (N - 2 N_u)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "The magnetizati on of the ensemble is given as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 30 "M = mu (N_u - N_d) = - U / B ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 147 "These relation s imply that the macrostate is specified by the total number N, the to tal energy U, or equivalently by specifying N and N_u (or N_d)." }} {PARA 0 "" 0 "" {TEXT -1 107 "The multiplicity of such a macrostate is given by the number of possibilities to chose N_u out of N states." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "restart; with(plots): ladd:= (L,e)->[op(L),e];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Omega:=(N,N_u) ->binomial(N,N_u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%laddGR6$%\"LG %\"eG6\"6$%)operatorG%&arrowGF)7$-%#opG6#9$9%F)F)F)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%&OmegaG%)binomialG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "The number of spins" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "N:=100;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"$+ \"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 264 "The energy range is from - 100 to 100 (or -N to N) given that the spins can all be aligned up to \+ counteraligned. First we calculate the energies, multiplicities and en tropy: given that a single spin can only be aligne or counteraligned t he loop runs over the spins:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "U1:=-[]: O1:=[]: S1:=[]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "for N_u from N to 0 by -1 do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "u1:=(N-2*N_u);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "o1:=Omega(N ,N_u);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "s1:=log(o1);" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 50 "U1:=ladd(U1,u1); O1:=ladd(O1,o1); S1:=ladd(S 1,s1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 3 "U1;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7aq!$+\" !#)*!#'*!#%*!##*!#!*!#))!#')!#%)!##)!#!)!#y!#w!#u!#s!#q!#o!#m!#k!#i!#g !#e!#c!#a!#_!#]!#[!#Y!#W!#U!#S!#Q!#O!#M!#K!#I!#G!#E!#C!#A!#?!#=!#;!#9! #7!#5!\")!\"'!\"%!\"#\"\"!\"\"#\"\"%\"\"'\"\")\"#5\"#7\"#9\"#;\"#=\"#? \"#A\"#C\"#E\"#G\"#I\"#K\"#M\"#O\"#Q\"#S\"#U\"#W\"#Y\"#[\"#]\"#_\"#a\" #c\"#e\"#g\"#i\"#k\"#m\"#o\"#q\"#s\"#u\"#w\"#y\"#!)\"##)\"#%)\"#')\"#) )\"#!*\"##*\"#%*\"#'*\"#)*\"$+\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot([[U1[j],S1[j]] $j=1..N+1]);" }}{PARA 13 "" 1 "" {GLPLOT2D 743 227 227 {PLOTDATA 2 "6%-%'CURVESG6$7aq7$$!$+\"\"\"!F*7$$ !#)*F*$\"1#4))f=q^g%!#:7$$!#'*F*$\"1NFcbG92&)F07$$!#%*F*$\"1?lb/)\\$*> \"!#97$$!##*F*$\"1p[Hm9>=:F;7$$!#!*F*$\"1V#)>%\\#o8=F;7$$!#))F*$\"1\" \\NkB%*)*3#F;7$$!#')F*$\"1gpv*pK'\\BF;7$$!#%)F*$\"1-V!\\\\[\\f#F;7$$!# #)F*$\"1%ev[*[SFGF;7$$!#!)F*$\"1lyAOBB[IF;7$$!#yF*$\"1a5)fxB%eKF;7$$!# wF*$\"1oa(zu'zeMF;7$$!#uF*$\"1Nrn$\\N+l$F;7$$!#sF*$\"1o5es0sKQF;7$$!#q F*$\"1)>'4#G]t+%F;7$$!#oF*$\"1^7_NlNuTF;7$$!#mF*$\"1h***4)o6MVF;7$$!#k F*$\"1/+*fwjp[%F;7$$!#iF*$\"1&y*z#z\">LYF;7$$!#gF*$\"1I;\"4[jIx%F;7$$! 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To check the dimensions one should put the right \+ units back in. Where entropy is shown, it is really S/k; where the hea t capacity is shown, it is C/k, where the temperature is shown, it is \+ kT/e for the Einstein solid, and kT/(mu B) for the paramagnet, etc." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "122 1" 0 } {VIEWOPTS 1 1 0 1 1 1803 }