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# Quant Chem & Stat Thermodyn II CEM 992

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This 9 page Class Notes was uploaded by Ladarius Rohan on Saturday September 19, 2015. The Class Notes belongs to CEM 992 at Michigan State University taught by Staff in Fall. Since its upload, it has received 56 views. For similar materials see /class/207704/cem-992-michigan-state-university in Chemistry at Michigan State University.

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Date Created: 09/19/15

Michigan State University Chemistry 992 Lecture 7 Thermodynamic Connection for the Grand Canonical Ensemble De ne EltZVTgt22NQNVT GCE pa1tition function note N gt 00 Then by integrating p over F and summing overN r pV no N dFN r H no N e 22 We 2 Idl p l N 39 N Thus e pVEzVT 1 or pV 1nEzVT thermodynamic connection Thermodynamic formulas dA SdT pdVde Want uz as the independent variable Use a Legendre transformation duN A SdTpdVNdy Now recall Euler s theorem for degree one homogeneous functions uN E TS pV and the definition of Helmholtz free energy A E TS Thus uN A pV and dpV SdTpdVNdy Now we have pV pVuVT Sincey uz the pV product depends on the desired independent variables Then CEM 992 7 Lecture 7 360 kT61iE klnE V T Vz pkfal j kT 6V 7 V The last follows from the de nition of E It also follows because uzandT are intensive variables Think about the last statement Also N kT 61 6 TV These are the three ensembles most used in Statistical Mechanics l Microcanonical 7 theory E W fixed isolated 2 Canonical 7 calculation TVN fixed thermal 3 Grand canonical 7 when fixed N is a problem as in QM and multi component T V u fixed thermal and material They correspond to dszerent physical situations Note that in thermodynamics we pick different independent variables for different physical situations and de ne different thermodynamic potentials e g A A NVT G G NpT This leads to the thought that different ensembles correspond to different thermodynamic functions How do we find the thermodynamic functions for a given ensemble Why should this be true It is due to the smallness of uctuations This leads to the equivalence of ensembles and suggests that we use whichever ensemble is convenient for a particular calculation Let s fool around and see what happens SNVE k1noNIE klna and wrung But we may write Q as EJE Q IdPquNe Hlp q 2 9E 6 11 1 Nh CEM 992 7 Lecture 7 Just take regions of phase space of a given energy E We may also write EJE no 2 QE e pE ImEe39 EdE 11 0 where Q aEalE is number of states of energy E about 5E aE is a density of states at energy E Thus M klnIdEwEe39pE Also we have Z VklnEuVTkln o e Ndee39pEaE N0 0 A pattern emerges We keep integrating summing over extensive variables with weight e39 to get the intensive variable Try for example E V dee E e pyaNVEdV YNTP 0 0 What s the characteristic thermodynamic potential G GNTP You show in the problem set General method for obtaining thermodynamic function Replace Q by eSk in the partition function and sum exponents See McQuarrie 315 e g E i nTS E pV A pV G Y k kT kT recallpVuN ETS and GApV Thermodynamics can use any independent variables and then de ne a thermodynamic potential Statistical Mechanics can de ne an ensemble to match any choice of independent variables CEM 992 7 Lecture 7 Why does the above summation trick work Dominance of maximum term That is IdEe39 EIdVe39ppVaE e39pEe39 pVaE EVTS e p ZeG Thus because the mean and the most probable values are the same we can obtain any thermodynamic function from any ensemble Thus all the ensembles are equivalent Conclude 1 all ensembles give the same result 2 can de ne ensembles by above recipe Fluctuations 1 If the uctuations were not small then F couldn t be identi ed with thermodynamics This shows that 2 The ensembles are equivalent and 3 Provides useful information Fluctuations in the Canonical EnsembleiT fixed E uctuates Look at McQuarrie for quantum mechanical analog Zinstead of J Know how to do both E H WLNJJ39dFNH e pHQ Write zero as 1 j dr E HeEAW H 0 Now differentiate the above with respect to T N 5E L Armw 0 jdr 6TE H kT2A HkT 6Te jdr a E LE HZe HkT CV k ltE H2gt l kT2 where we combined terms as A H ST and recalledA TS E Therefore kTZ CV ltH2 H 2 Michigan State University CEM 992 Lecture 22 Linear Response Theory Formal Ohm 3 Law is a Linear Law 1 O39E The current is the product of the coefficient of conductivity that is independent of the eld E and the field J is the response to stimulus E The proportionality constant 6 is only a property of the medium Compare this statement with our previous result on IR absorption More generally 10 J39jwd3t t39Ft39dt39 d3 t is known as a res onse unctz39on In the linear a roximation it is 39ust a 3 Stem P PP J y function Change variables to write 1 J t39Ft t39dt39 3 Ja XaFa where A m I A te39 quotdt is the onesided Fourier transform Laplace see Forster book Take Ft39 5t39 t0 Then J t t t0 ltJt0 for tltt0 Causality d3 I is the response to a deltafunction disturbance We want a molecular expression for d3 I Write the Hamiltonian for the system plus an interaction with an external field F Take HT H H39t tdependence of H39t 3 mixedclass QM H is system Hamiltonian H 39t is interaction with external field For example dipole coupled to electric field CEM 992 7 Lecture 22 Calculate the response of the system density matrix to the external eld Then average to nd B to terms linear in external eld This is why it s called linear response theory Take H39t BFt Ft is a cnumber B is an operator 61 6t HT p7 Schrodinger representation 6g Hp7FtB37Il t Then p7 t ptJ foo Ut t39Ft39BpT U t t39dt39 where Ut aim Ut is system evolution Thus pt Utp0 has no F t dependence This is an exact result that you should verify by differentiating with respect to t to regenerate the differential equation Iterate in F to terms linear in F pTt ptJOUt t39Ft39lBpt39U HM 0F2 BT t BtJo Tr BUi i39B pt39U i t39Ft39 dt where ltBgtltrgtTrBpltrgt Now in the absence of the external perturbation the system is in equilibrium by assumption Thus ptpi CEM 992 7 Lecture 22 Also note that we may choose 32 0 since if not de ne B39B ltBZ so that 3 0 With these assumptions i t 39 r r ltBgtTg wTrBUt tBpU t tFtdt The system response to F involves only the equilibrium density matrix p But note the t dependence through U I Manipulate Let t t39 139 define BH I Bt UlBU to obtain BT JltBrBlFt rdr Compare with the Black Box expression that we started with to obtain the response function expression ltrgtiTrpBltrgtBiEgltBltrgtBl The response function is the analog of the partition function in equilibrium statistical mechanics It connects macroscopic to microscopic You now know what you need to calculate though it may be hard Remember this is a special nonequilibrium state small deviation from equilibrium Some properties G130 413 0 Proof Consider Note A E 140 A to At to Timetranslational invariance take to t therefore ltAlt rgtAgtltAAltrgt lt1 and if we let I gt t CEM 992 7 Lecture 22 Thus t t Now consider the onesided Fourier transform 1a I tequot dt l 1quotw1mzw Emu 2a Therefore after using the time symmetry above and ipping the integration limits you show that u 1 7 I m g M ltBBtle dt Alternate forms 1quota2 hl e39 h LO BB e39mdt Show this by expanding in an energy basis quot 1 WW 7 z w Etanh 2 NM Bt31e ddt 61 b ab ba anticommutator Let us calculate energy dissipation into the system The external eld F induces transitions in the system both absorption and emission But some gets degraded into heat by Vibrationalrotationaltranslational exchanges CEM 992 7 Lecture 22 Evaluate f and average over at least one cycle of external eld of frequency 03 according to i 2 dt36F 27239 0 6t De ne the cycleaverage Qa ltgt f Then using the expression for H one has noting that Ft Fm e 2 equot F is monochromatic and real ia a Qw7ZW ZWTIFJZ 31 amt Thus 9agrantflt3ltrgt31e drIFr This is the uctuationdissipation theorem We can obtain dissipation into the system from the equilibrium time correlation function TCF 1 cltrgt 3lttBltrt31gt The TCF represents the uctuation of B away from its equilibrium value Postulated by Lars Onsager A spontaneous uctuation in the system decays on the average in the same way as an externally produced disturbance decays if the deviation from equilibrium is not too great The TCF plays the role as the PF in equilibrium theory It tells you what you must calculate though it may be hard to evaluate Note that BtUltBUt and Ute7m has full manybody system Hamiltonian

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