The root mean square value of the distribution is the square root of the mean square value of x: 1/2. Xrms

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1 Background and Rfrnc Matral Probablty and Statstcs Probablty Dstrbuton P(X) s a robablty dstrbuton for obsrvng a valu X n a data st of multl obsrvatons. It can dscrb thr a dscrt ( = 1 to N) data st or contnuous functon. For th contnuous dstrbuton, P(X) s known as a robablty dnsty, and P(X) dx s th robablty of obsrvng a valu btwn X and X + dx wthn a larg saml. ll robablty dstrbutons ar normalzd such that Dscrt Contnuous N P 1 PXdX 1 1 h man or avrag valu of X for ths dstrbuton s 1 N N N 1 1 X X PX X P X dx h root man squar valu of th dstrbuton s th squar root of th man squar valu of x: Xrms X 1/ h man s th frst momnt of th dstrbuton, th man squar valu s th scond momnt, and th n th momnt of P(x) s n 1 N n N n n N 1 1 X X PX X P X dx h wdth of th dstrbuton functon about ts man s quantfd through th standard dvaton σ or varanc σ : X X Gaussan Dstrbuton h Gaussan normal dstrbuton s a contnuous dstrbuton functon commonly usd to dscrb a larg numbr of ndndnt samls from a random dstrbuton. P X 1 ( X X) x h dstrbuton functon s comltly dscrbd by ts man and varanc. ll hghr momnts of ths dstrbuton ar zro. h most robabl valu, or ak, of th dstrbuton s th man, and th wdth s gvn by th standard dvaton. h robablty dnsty wth ±1σ of th man s 68.% ndr okmakoff 6/9/17

2 and wthn ±σ s 95.4%. h full-wdth of P(X) at half th maxmum hght (FWHM) s 8ln.355. Bnomal and Posson Dstrbutons bnomal dstrbuton s a dscrt robablty dstrbuton that rlats to th numbr of tms of obsrvng N succssful vnts from a srs of M ndndnt trals on a bnary varabl (ys/no; u/down). For nstanc th numbr of tms of obsrvng N cos of th amno acd tyrosn wthn a rotn squnc of lngth M. ltrnatvly w can say that t gvs th robablty that N random vnts han durng a gvn rod of tm, M. Lk th numbr of barrr crossngs (ums) durng a tm wndow n a two-stat tractory. Consdr th numbr of ways of uttng N ndstngushabl obcts nto M quvalnt boxs: M! M N!( M N)! N (Not: In th lmt M N, Ω M N /N!). h chocs mad hr ar bnary, snc any gvn box can thr b mty or occud. Ω s also known as th bnomal coffcnt. Howvr, unlk a con toss, a bnary varabl nd not hav qual robablty for th two outcoms. W can say that w know th robablty that a box s occud s o. hn th robablty that a box s mty s = 1 o. h robablty that n a gvn ralzaton wth M boxs that N ar occud and M N ar N M N mty s. hn multlyng th robablty for a gvn ralzaton and th numbr of o ossbl ralzatons, th robablty of obsrvng N of M boxs occud s M M! PNM (, ) o o (1 o) N N!( M N)! N M N N MN hs s th bnomal dstrbuton. h avrag numbr of occud boxs s N = N o. Gaussan dstrbuton mrgs from th bnomal dstrbuton n th lmt that th numbr of trals bcoms larg (M ). h man valu of th dstrbuton s No and standard dvaton s No(1 o). Posson dstrbuton mrgs n th lmt that th numbr or trals (M) bcoms larg and th robablty of obsrvng an occud box s small, o 1. N N PN ( ) N! N

3 h man of th Posson dstrbuton s N and th standard dvaton σ = N 1/. Fluctuatons n 1/ N scal as σ, and th fluctuatons rlatv to th man as / NN. hn w s that fluctuatons ar most aarnt to th obsrvr for small N wth th bggst contrast for N = M. B. Jackson, Molcular and Cllular Bohyscs. (Cambrdg Unvrsty Prss, Cambrdg, 6), Ch. 1. 3

4 Constant and Unts Practcal Unts and Idntts for Bohyscal Puross 1 m 1 9 nm 1 1 Å 1 1 m 1 nm 1 9 m 1 N 1 1 N 1 aj 1 18 J zj 1 1 J 3 N nm 3 zj nm 4 t 3 K k B 4.1 N nm.5 kj/mol.6 kcal/mol 1 56 nm 4 k B k B 5 mv 4

5 hrmodynamcs Frst Law du dq dw w: Work rformd by th surroundngs on th systm. Mchancal work: lnar dslacmnt, strtchng surfac aganst surfac tnson, and volum xanson aganst an xtrnal rssur. Elctrcal work: dw qe dr Magntc work: dw s ntrnal to th systm. q: Hat addd to th systm. dw f dr da dv xt xt M db s fld s xtrnal to th systm or dw B dm f th fld C: Hat caacty lnks hat and tmratur. t constant rssur: dq Cd Scond Law ds dq rv Stat Functons Intrnal Enrgy, U: U q w du ds dv dn Natural varabls: N, S, V Enthaly, H: H U V dh ds Vd dn Natural varabls: N, S, 1 H ( ) H ( ) C d Hlmholtz fr nrgy, or F: U S d dv Sd dn Natural varabls: N, V, mor gnrally: d S d wrv whr wrv dv dn f dr da dq Gbbs fr nrgy, G: G H S V dg Sd Vd dn Natural varabls: N,, mor gnrally: dg Sd wrv 1 5

6 whr w rv Vd dn f dr da dq For a rvrsbl rocss at constant and, ΔG s qual to th non-v work don on th systm by th surroundngs. Gnrally: (dg), w non-v. Chmcal otntal: Entroy: G N N,,{ N }, V,{ N} G S V If you know th fr nrgy, you know vrythng! V G V H G S H G G U H V U G G U S G G C S C G G C V U Sontanous Procsss Condtons that dtrmn th drcton of sontanous chang and th condtons for qulbrum dg d ds du dh,, N V,, N H, V, S, N, S, N Chmcal Equlbra G rxn H rxn K q x(g rxn S rxn / R ) G rxn roducts N N ractants 6

7 Statstcal hrmodynamcs h artton functons lay a cntral rol n statstcal mchancs. ll th thrmodynamc functons can b calculatd from thm. Mcrocanoncal Ensmbl (N,V,E) ll mcrostats ar qually robabl: P = 1/Ω whr Ω s th numbr of dgnrat mcrostats. S kln Boltzmann ntroy n trms of dgnracy Canoncal Ensmbl (N,V,) vrag nrgy E s fxd. Canoncal Partton Functon: Q For on artcl n n dmnsons: Q E / k E/ k de 1 H ( q, )/ k n n n Q dq d h h classcal artton functon for N non-ntractng artcls n 3D: 1 Q d d 3 N H(, q )/ k h, N whr,, q q q q q x y z 3 If th kntc and otntal nrgy trms n th Hamltonan ar sarabl as N V q thn h / mk. So whr 1/ 3 3 ( )/ 1 N N V q k 3 N ( )/ k Q dq d h 3 N 1 V q Q dq / k, 7

8 Mcrostat Probablts P E / Q E k or dscrbng th robablty of occuyng mcrostats of an nrgy E, whch hav a dstngushabl dgnracy of g(e): P E E / k ge ( ) Q or as a robablty dnsty: Pr E( r)/ k Q Intrnal Enrgy Hlmholtz Fr Enrgy U E P E 1 Q E E /k Q(N,,V) lads to (N,,V). Usng d dv Sd dn : U S U k lnq 8

9 ll th othr functons follow from Q, U and U 1 U k C V U lnq S U C S V N k lnq k lnq,n,v k k lnq V lnq N,N,V k Entroy Entroy n trms of mcrostat robablts and dgnracs S k U k P E k But E k lne k, so S k P ln E k lnq k P ln E Q P ln P S k P ln P lnq Gbbs quaton: S n trms of mcrostat robablts 9

10 Ensmbl vrags Othr ntrnal varabls (X) can b statstcally dscrbd by N X PX P E 1 Q E / k For our uross, w wll s that translatonal, rotatonal, and conformatonal dgrs of frdom ar sarabl Q Q trans Q rot Q conf... Grand Canoncal Ensmbl (μ, V, ) vrag nrgy E and avrag artcl numbr N ar fxd. hrmodynamc quantts dnd on mcrostats () and artcl ty (). P ( ) s th robablty that a artcl of ty wll occuy a mcrostat of nrgy E: P ( ) E ( )/k B N ( )/k B U EP ( )E ( ) and N P ( ) N( ) h grand canoncal artton functon s Q(N,V, ) N ( )/k B Strlng s roxmaton For larg N: N! N ln N N or N! N N N N! N ln N N 1 ln( N ) Frst Law Rvstd Lt s rlat work and th acton of a forc to changs n statstcal thrmodynamc varabls: h ntrnal nrgy:.. L. Hll, n Introducton to Statstcal hrmodynamcs. (ddson-wsly, Radng, M, 196), ,

11 U E P E du dee dp P de Not th rlatonsh btwn ths xrsson and th Frst Law: du d q d w d q rv ds E dp dw dv P de 11

12 Contnuum Elctrostatcs h ntracton of chargs can b formulatd through a Forc, Fld, or Potntal. Consdr th ntracton btwn two ons and B, saratd by a dstanc r, wth chargs q and qb. q r qb B Forc Coulomb s Law gvs th forc that B xrts on. r ( r r ) r B r rˆ r / r f 1 qq 4 B r rˆ 1 Work Elctrcal work coms from movng chargs dw f dr s long as q and ar ndndnt of r, and th rocss s rvrsbl, thn work only dnds on r, and s ndndnt of ath. o mov artcl B from ont 1 at a saraton r to ont at a saraton r rqurs th followng work w qqb 4 r r q r r qb Fld, E h lctrc fld s a vctor quantty that dscrbs th acton of charg B at ont s E ( r ) 1 q rˆ B 4 r rˆ s a unt vctor ontng from r B to r. E s rlatd to forc that chargd artcl B xrts on a chargd tst artcl at through f q E ( r ) Mor gnrally for th fld xrtd by multl chargd artcls at ont r s th vctor sum of th fld from multl chargs (): 1 q E( r ) E ( r ) rˆ 4 r 1

13 whr r r r and th unt vctor rˆ ( r r )/ r. ltrnatvly for a contnuum charg dnsty ρ(r), 1 ( r r) E( r) ( ) d 3 4 r r r r Elctrcal work coms from movng chargs n an lctrostatc fld s dw qedr Elctrostatc Potntal, Φ h lctrostatc otntal Φ s a scalar quantty dfnd n trms of th lctrc fld through E or E r dr h ntgral xrsson llustrats th rlatonsh btwn th lctrostatc otntal at ont and th work ndd to mov a artcl wthn that otntal. h lctrostatc otntal at ont, whch rsults from a ont charg at B, s 1 q 4 r B or for a contnuum: 1 ( r) ( r) d 4 r r r s rlatd to th otntal nrgy for ntracton btwn two chargs as 1 qq U q qbb 4 r hs s th Coulomb ntracton otntal. Mor gnrally, for many chargs, th ntracton nrgy s 1 U q( r) 1 ( ) ( ) d r r r h factor of ½ s to assur that on dos not doubl count th ntracton btwn two artcls. B 13