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# GRAD PROB. SOLVING PHY 6938

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This 8 page Class Notes was uploaded by Garett Kovacek on Thursday September 17, 2015. The Class Notes belongs to PHY 6938 at Florida State University taught by Staff in Fall. Since its upload, it has received 45 views. For similar materials see /class/205544/phy-6938-florida-state-university in Physics 2 at Florida State University.

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Date Created: 09/17/15

Statistical Thermodynamics review In this course we will not provide a comprehensive introduction to elementary thermodynamics and statistical mechanics which have already been covered in PH Y 5524 It is nevertheless useful to briefly review the basic ideas and results form these subjects which will serve as the starting point for the more advanced topics we will study in detail PHENOMENOLOGICAL THERMODYNAMICS Thermodynamics has historically emerged much before its microscopic basis has been established It describes the basic laws of thermal behavior as directly observed in experi ment Remember the early steam engines were built much before Boltzmann s discoveries In many novel materials we still do not have a well understood microscopic theory but thermodynamic laws certainly apply and they are useful in describing and interpreting the experimental data We Will therefore pause to refresh our memory of elementary thermodynamics A more detailed discussion can be found7 for example7 in the short but beautiful text by Enrico Fermi We concentrate as Fermi does on a given example of a PVT system liquid or gas The results are then easily generalized7 for example7 to magnetic or other systems as well The First Law In equilibrium thermodynamics7 one considers so called 77reversible77 processes7 where the physical state of the system is changed very slowly7 in tiny7 in nitesimal steps The First Law is simply a statement of energy conservation It states that the in nitesimal change of the internal energy E of the system is the sum of the work W done against an external force and the heat ow Q into the system dE dQ 7 dW For example7 when a gas is expanding7 dW pdV7 where p is the pressure of the gas7 and dV is the volume change of the container Heat capacity at constant volume if the gas is kept at the same volume7 but is heated7 then dE dQ7 and the constant volume heat capacity is de ned as d 6E CV 2 7 dT V 6T V Heat capacity at constant pressure If we x the pressure then V is a function of V and T7 it is easy to show Problem 1 that dQ 6E 6V OFF i p dT p 6T P 6T p Magnetic systems In magnetic systems7 one usually considers the internal energy as an explicit function of the external magnetic eld h If the external eld is in nitesimally varied7 then the work done is dW iMdh where M is the magnetization of the sample We can write dU dQ Mdh The Second Law The 77problem77 with the rst law by itself is that it does not tell us how much heat ows in or out of the system It is only useful if we have a thermally isolated system dQ 07 or if we already know the equation of state ie the form of VTp The essential content of the Second law is that systems left alone tend to assume a most probable state7 ie the one where as a function of time it explores as many con gurations as allowed by energy conservation It can be formulated in many ways7 which can be shown to be mathematically equivalent see book by Fermi For example7the formulation by Clausius says Heat cannot spontaneously flow from a colder to a hotter body But its most important consequence is that it introduces the concept of entropy For any reversible process7 the change of entropy is given by dQ dS 7 T If the system is thermally isolated dQ 07 then any process reversible taking the system from state A to state B results in no change of entropy However7 based on the Second Law7 one can show that if an isolated irreversible process is considered7 then strictly SB 2 SA In other words7 systems left to themselves tend to equilibrate by strictly increasing their entropy The First Law for a gas can now be written as dE TdS i pdV From this expression7 we can write T LE CLE as V p ov S39 The Third Law Nernst Theorem For classical systems the entropy is de ned up to a reference constant as it is not clear how degenerate is the ground state In quantum mechanics though quantum tunneling tends to lift the ground state degeneracy and in most systems the ground state is not degenerate Therefore the entropy of this state vanishes ie This result allows one to explicitly determine the precise numerical value of the entropy at any temperature directly from experimental data as follows We can express the entropy change in terms of the speci c heat and write Tom ST TdT 0 Note that this expression immediately shows that CT has to vanish at T 0 otherwise the integral would diverge This expression is often used in interpreting experiments for example on spin systems Since the entropy reaches its maximal value for noninteracting spins by looking at the temperature where ST starts to saturate we can estimate the energy scale of the spin spin interactions The Free Energy The internal energy must be regarded as an explicit function of the volume V and the entropy S as independent variables However this form is not particularly convenient to use since we cannot directly control the entropy in an experiment It is often more convenient to consider temperature T and the volume V as independent variables To obtain an expression similar to the First Law except with T and V as independent variables we de ne the quantity F E 7 TS called the free energy Mathematically the free energy can be regarded as a 77 Legendre transform77 of the internal energy and its total differential can be computed using the chain rule7 as follows dF dE 7 TdS 7 SdT TdS i pdV i TdS i SdT OI dF iSdT i pdV Important result 77If the free energy is known as a function of its natural variables T and V then from it all other thermodynamic quantities can be computed 5 LF LF oTV p oVT39 We can also get the internal energy Ha z V For example 6T 7 dQ i as 7 NF OV WV ToV TWV39 For magnetic systems and the speci c heat dF iSdT Mdh 6F M a and the magnetic susceptibility 7 3M 7 62F X oh T th T39 Stability conditions and we get the magnetization Consider a gas at temperature To and volume V Now let us assume that the volume of the system is suddenly increased and the system allowed to relax The gas will rapidly expand7 but this will correspond to an irreversible process Assuming that the gas continues to be in contact with a heat reservoir at temperature To7 some heat AQ gt 0 must ow into the system This is true7 since without thermal contact the gas would simply cool down by adiabatically expanding If the heat contact is there then the gas will re heat by absorbing some heat from the reservoir Note that this is consistent with the Second Law which demands that the entropy strictly increases in such an irreversible process But what happens to the free energy Well according to the rst law the change of internal energy AU AQ 7 AWbut since the volume expanded rapidly no work was actually done by the gas and thus AU AQ Now we note that even for an irreversible process the change of entropy is d 1 A ASZidQTOQ39 We conclude that AQ S TOAS for such an irreversible process As a result the change of the free energy AF AEiTOAS 0 We conclude that if the system is mechanically isolated so no mechanical work is done in a given irreversible process then the free energy of the system cannot increase This argument is very general and can be easily repeated for any thermodynamic system Therefore The free energy is at a minimum in the state of stable equilibrium This result is very important since we often resort to minimizing of the free energy with respect to some order parameter in order to identify the thermodynamically stable states of the system These stability conditions stating that at the equilibrium point the entropy is at a max imum and the free energy at a minimum lead to few other important results It is possible to show see Problem 2 that it leads to the following conditions for the speci c heat CV the isothermal compressibility HT and the isoentropic compressibility H5 valid in the equilibrium state 65 6V 6V 7 gt 7 7 gt 7 7 gt 0v TaTW math 622 All these results are valid for systems with xed numbers of particles N Of course these expressions are easy to generalized when instead the chemical potential LF 6N Ty is considered xed eg if we have a particle reservoir In this case we perform a Legendre transform with respect to N de ning the so called 77Grand li otential77 QF7MN we 3 T MICROSCOPIC APPROACH so that for example Statistical mechanics as developed by Boltzmann Gibbs and others has provided the microscopic basis for thermodynamics In this brief review we will not repeat the discussions relating to the de nitions of the various ensembles or the derivations for the expression for the partition function The central result that we will use over and over is the expression for the free energy in terms of the partition function 76F ln Z where B T l Here and in the following we will use units of energy where the Boltzmann constant k3 1 The partition function generally takes the form Z Z expe En where En are the energy states of the system Out main task is to develop strategies how the partition function can be calculated We end this brief summary of Statistical Thermodynamics with a few comments about the microscopic de nition of the entropy following the discussion form Kadanoff Chap 8 For a closed system microcanonical ensemble with energy E the entropy is de ned in terms of the density of energy states expsltEgt 26w 7 En Such a de nition of the entropy is motivated by the fact that the number density of energy states generally grows exponentially with the number of degrees of freedom N while the entropy must be extensive ie proportional with N For this function to be a smooth analytic function of energy one has to consider the thermodynamic limit N a 00 where any discrete spectrum turns into a continuum one Note that for systems with a nite number of degrees of freedom and a discrete spectrum eg a single quantum spin7 this de nition of SE does not make sense Physically7 we can say that the entropy measures the density of accessible states at a given energy Now we can see the microscopic basis for the Second Law in equilibrium all accessible states of a given energy tend to be equally populated7 maximising the entropy Using this de nition7 we can rewrite the expression for the partition function as Z deexpS6expi e it is interesting to examine this integral in the thermodynamic limit N 6 00 Since 56 6 0 at 8 6 07 the integrand is dominated by a sharp peak at some 8 E 7 which becomes increasingly sharper and sharper as N 6 007 and the integral can be evaluated by a steepest descent method To determine 8 E 7 we look for the maximum of the integrand7 and we nd 358 38 5E We thus recover the relation between the microcanonical entropy and the temperature To leading order large N7 the partition function reduces to the integrand evaluated at the saddle point 26 expSE 61376 exp F the expected relation between the free energy and the entropy

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