Various Ensembles in Statistical Mechanics and the Derivation of Thermodynamic Properties

Various ensembles in statistical mechanics and the derivation of thermodynamic properties.

To make the connection between statistical mechanics and thermodynamics, average quantities are calculated over a (very large) ensemble of individual systems. The ensemble itself is always a so-called isolated entity. It contains a specific total number of particles, it has a precise total volume and a precise total energy. In what follows each system comprising the ensemble is assumed to be in a particular energy eigenstate y j ,

and has a particular number of particles N j , a specific energy E j and a specific volume

Vj . The possible energies of a particular system are not completely arbitrary: Given the

volume and number of particles in the system, E j is an eigenvalue of the Schrödinger

equation, and depends on N j and Vj . Ej= E j( N j , V j ) . The states y j(N j , V j ) form a

ˆ linearly independent basis of energy eigenstates of the Hamiltonian H( Nj , V j ) . This

dependence on Vj, N j is implicit in what follows. The label j summarizes all of the characteristics of an individual system and it means that it corresponds to a particular

Vj, E j , N j ,y j . Specifying j defines everything, and this is referred as the “state of the system”. The basic unknowns are the probabilities to find a system in a particular state. These probabilities are provided by the partition function. In what follows below, a (general) partition function O, depending on variables x, y ,..., is defined as a sum over relative probabilities O(x, y,...) = P (x, y,...) å j (1) j and the associated normalized probabilities to find a system of type j (or in state j) in the ensemble are then given by P j (2) Pj (x, y,...) = (x, y,...) O Knowledge of the probabilities allows one to calculate ensemble averages, notably

1 Ej P j = E = U j N P= N = N j j (3) j

Vj P j = V = V j In addition the probabilities define the entropy of any type of ensemble as -k Pln P = S j j (4) j The above provides a recipe to calculate ensemble averages, and thermodynamical quantities. It is not implied that each system in an ensemble is indeed described by an eigenfunction of the Hamiltonian. The argument is that if we make this assumption, and calculate the averages in the prescribed fashion, we obtain agreement with the laws of thermodynamics. The understanding of the precise physical nature of a large number of molecules in accordance with time-dependent quantum mechanics is a non-trivial problem. One would like to be able demonstrate that the wave function of a large system of interacting molecules (in the gas phase for example) evolves in time such that average values of molecular quantities follow the laws of statistical mechanics, and quickly become more or less independent of time. On the most fundamental level statistical mechanics would be expected to derive from quantum mechanics and the time-dependent Schrödinger equation. There is nothing in the time-dependent Schrödinger equation to suggest that systems are to be eigenstates of the Hamiltonian. Rather, the argument is made that thermodynamic properties, for systems in equilibrium, are independent of time. This is achieved by taking systems to be described by stationary states, i.e. eigenfunctions of the Hamiltonian, and taking an average to obtain ensemble properties. It will be clear that the properties calculated in this way, indeed will be independent of time, even if we would evolve the ensemble in time. In reality, systems do fluctuate and show a time- dependence. For most intents and purposes their thermodynamic properties are independent of time, however, and this is described by the ensembles we will consider. It is good to point out that in an actual experiment we have one system, and this system itself attains thermodynamic equilibrium. In statistical mechanics the system is replicated many times, and we calculate averages over the replicas, assuming they are each described by an eigenstate of the Hamiltonian. This representation of the situation is

2 clearly quite different from the actual situation, and it is a bit of a miracle why this all works. Let us leave these (poorly understood) fundamental questions behind and return to the derivation of thermodynamic properties along the conventional lines of statistical mechanics.

The ensemble can consist of isolated systems, meaning the volume, energy and number of particles is the same for each element in the ensemble, Vj= V, N j = N , E j = U " j . This is called the microcanonical ensemble. Another widely used ensemble is the canonical ensemble, in which each system is closed, meaning it is allowed to exchange energy, but not matter with neighbouring systems. In a canonical ensemble each system has the same number of particles Nj = N and volume Vj = V , but the energy is specific

for each state and is denoted as E j . This is the ensemble discussed in Metiu. In the grand

canonical ensemble the individual systems can differ in both the number of particles N j

and their energies E j , but the volume is fixed, Vj = V . We will discuss two more ensembles, one in which only the number of particles is fixed, while the volume and energy can vary. In the generalized ensemble, all extensive variables, Vj, N j , E j can vary.

The partition functions either depend on the constant=average value for V, N and/or U, or they depend on an intensive variable that is a Lagrange multiplier associated with the constraint that the total number of particles, the total volume or the total energy of the complete ensemble is constant. The thermodynamic variable conjugate to preserving total energy is the temperature T. The variable associated with preserving the total volume is the pressure p, while the variable associated with the number of particles is the chemical potential m . Therefore, any partition function “O” has independent variables as follows O( U orT , V or p , N or m ) , (5) with the actual choice of variables depending on the extensive variables U, V, N that are kept constant in each system in the ensemble. The partition functions then relate to a thermodynamic potential that has precisely the same natural variables as the ensemble, for example, we have seen already A( T , V , N )= - kT ln Q ( T , V , N ) . We will find other

3 similar relations. From thermodynamics we know that the Helmholtz free energy is the suitable thermodynamical potential to consider when a system is kept at constant temperature through a reservoir. Here the idea is similar. The other systems in the ensemble act as the heat reservoir, allowing a redistribution of energy, and providing a resulting partition function that depends on the temperature T, that might be viewed as a controllable parameter, or, equivalently, a variable in the partition function. The probabilities of the most likely distribution depend on the value of T, which was used as a Lagrange multiplier required to keep the total energy constant (as would be the case when we consider a system in contact with a heat reservoir: the total energy of system + reservoir would remain constant). We will see that the various types of ensembles we can create map precisely to the kind of thermodynamic potentials that are generated by Legendre transformations in thermodynamics.

Thermodynamic properties can be derived from any of these ensembles, and the final results are equivalent. For example, Metiu discusses the results for the canonical ensemble and this provides all thermodynamic properties. This feature of statistical mechanics is reflected in the fact that the thermodynamic potentials, when viewed as functions of their proper ‘natural variables’ all yield complete thermodynamic information. We will discuss the derivations in a unified context for the various ensembles, as it puts the theory in a general framework. It more clearly shows what is involved, and what freedoms exist to derive the results. Moreover, to derive certain results in statistical mechanics it may be far more convenient to use a particular ensemble, as the mathematics is ‘easy’, or even feasible, only for certain ensembles. So it is good to know about the existence of various ensembles. They are part of the tricks of the trade. The fact that various ensembles in statistical mechanics lead to basically identical results hinges from a physical perspective on the fact that even if one allows volumes, particle numbers and energies to vary per system, the fluctuations around the mean are very small, for large enough individual systems. This is well known from experience. For example, we expect temperature, pressure and density only to vary very little in a macroscopic system in equilibrium.

4 Let me sketch the basic procedure for any type of ensemble, and provide the knowledge that we assume as input. This summary will be somewhat abstract at first reading, but as we go through examples, I think this summary may prove useful to you. In the derivations below I assume that entropy is given by the basic formula S= - k Pln P j j (6) j The quantities temperature, pressure and chemical potential are defined through the partial derivatives of entropy, as it was done in the notes on thermodynamics: hence, 骣抖S1 骣 S p 骣 S m 琪汉 , 琪 , 琪 � (7) 桫抖UV, N T 桫 V U , N T 桫 N U , V T We will restrict ourselves to one component systems here. For all of the ensembles the procedure is that we maximize entropy subject to the constraints imposed on the ensemble

Ej= U " j or E j P j = U + Lagrangemultiplier b j V= V " j or V P = V + Lagrangemultipliera j j j (8) j

Nj= N " j or N j P j = N + Lagrangemultiplier g j

The constraints indicate that we can create 23 = 8 different ensembles in principle. From the maximization procedure for the constrained entropy we will find an expression for the partition function of the type Q = P ; P = P / Q å j j j , (9) j where the P will be simple exponential factors that correspond to unnormalized j probabilities. The logarithm of Q will be found to be related to particular chemical potentials that have the same natural variables as the variables in Q. We will find that maximizing the entropy subject to constraints is equivalent to minimizing (or sometimes maximizing) the chemical potential. We will see that from the procedure we will either obtain a relation between the Lagrange multiplier and intensive thermodynamic variables

5 1 p m (in particular: b=; a = , g = - ), or we will find an explicit expression for the kT kT kT intensive variables as an average over a mechanical variable, e.g.

骣抖Ej 骣 E j p= -琪 Pj, m = - 琪 P j (10) 邋抖 j桫VN j 桫 N V

Let us now discuss the various ensembles and establish the connections to thermodynamical quantities.

1. Microcanonical ensemble

In the microcanonical ensemble all systems defining the ensemble have an identical energy, volume and number of particles, each element of the ensemble is itself an isolated system. The partition function is simply the number of (linearly independent) quantum states, and is written as W(N , V , E ) (11) 1 The probability to find the system in a particular state is given by P = , j W(N , V , E ) which is the same for every state in the ensemble. This is precisely the fundamental postulate of statistical mechanics: In an isolated system each possible state is equally likely. The thermodynamic identification proceeds through Boltzmann’s fundamental law, S= kln W ( N , V , E ) (12) It is well known that in thermodynamics the condition for a spontaneous process in an isolated system is that entropy increases. Or: In a spontaneous process the logarithm of the corresponding partition function increases. The variables of the partition function are N, V, E, and these are also the natural variables of S in thermodynamics. Identifying E with U as usual, dU p m dS= + dV - dN(from dU = TdS - pdV + m dN ) (13) T T T Hence

6 骣禬ln 1 骣抖S 1 骣W骣抖 eln W 骣 ln W W 琪= 琪 =; 琪 =琪 = W 琪 = 抖抖桫EN, V k 桫 U N , V kT 桫 E N , V桫 EN, V 桫 E N , V kT 骣禬ln 1 骣抖S p 骣W p 琪= 琪 =; 琪 = ... = W (14) 桫抖VN, T k 桫 V N , T kT 桫 V N , T kT 骣禬ln 1 骣抖S m 骣W m 琪= 琪 = -; 琪 = ... = -W 桫抖NT, V k 桫 N T , V kT 桫 N T , V kT To unify the microcanonical ensemble with the treatment of other ensembles (see below), we can proceed alternatively as follows. The partition function for the microscopic ensemble is defined by finding the most likely distribution for the probabilities by maximizing -Pln P -l ( P - 1) 邋 j j j (15) j=1, W j = 1, W 1 This yields the partition function W(N , V , E ), P = . In the most likely j W(N , V , E ) distribution of the ensemble each state occurs equally likely. We can identify 1 1 S= kln W = k ln = - k邋 ln Pj = - k P j ln P j . (16) P W j j We will verify below that for every possible ensemble S= - k Pln P j j , (17) j and for every possible ensemble precisely this quantity is maximized under additional constraints depending on the particulars of the ensemble. At equilibrium the most likely distribution is reached (within fluctuations), and this is precisely what is meant by stating that entropy reaches a maximum at equilibrium. The (constrained) maximum of the

-k Pln P quantity j j defines the most likely distribution, and the most likely j distribution defines all thermodynamic quantities, as it is overwhelmingly more likely than any other distribution, and it is the only distribution that needs to be taken into account to define the average for the large ensembles under consideration.

7 2. Canonical ensemble.

In the canonical ensemble the individual systems in the ensemble can exchange energy, but they all have the same number of particles and volume. In thermodynamic language, each element of the ensemble is a closed system. The total energy of the ensemble is conserved (or a constant). This provides a Lagrange multiplier, b , associated with the constraint of energy conservation. The partition function is derived from the most likely distribution that preserves the average total energy U: i.e. maximize -Pln P -b ( P E - U ) - l ( P - 1) 邋 j j j j j , (18) j j j

P E= U, P = 1 from which one obtains 邋 j j j (stationarity w.r.t. band l ) j j Slightly more conveniently, we can maximize the unnormalized probabilities, and define the partition function accordingly. Hence maximize     å-Pj ln Pj - b(å Pj E j -U ) j j -bE (19)  j   ® Pj = e ; Q = å Pj ; Pj = Pj / Q j where we used the general result from the previous set of notes. Carrying out the maximization the partition function for the canonical ensemble is given by

-bEj - b E j Q( N , V ,b )= e ; Pj = e / Q (20) j where j runs over the possible states in the ensemble (see further notes, point 1, for further discussion). The expression for entropy is hence given by:

S�= k邋 Pjln P j - - k ( +b E j ) P j k ln Q P j j j j (21) =bkU + kln Q = b kU + k ln[ e-b E j ] j To establish the connection with thermodynamics consider the partial derivatives

8 1骣 S 1 =琪 = bk � b T桫 UN, V kT 骣抖E 骣 E 骣 E p骣 S 1j-b E j 1 j j =琪 =k邋 -b 琪 e = - 琪 Pj� p - 琪 P j (22) T桫抖 VU, N Qj桫 V T j 桫抖 V j 桫 V 骣抖E 骣 E 骣 E m 骣 S 1j-b E j 1 j j - =琪 =k邋 -b琪 e = - 琪 Pj� m 琪 P j T桫抖 NU, V Qj桫 N T j 桫抖 N j 桫 N Hence from the partial derivatives we obtain a definition for b , but we also find expressions for the intensive variables pressure and chemical potential as average values. Substituting the expression for b in the relation for entropy (Eqn. 21 ) we obtain S= U/ T + k ln Q � U= TS - kTln Q ( N , V , T ) A ( N , V , T ) (23) dA= dU - TdS - SdT = - SdT - pdV + m dN The natural variables for Q are N, V, T, and these are also the natural variables of the corresponding characteristic function A, the Helmholtz free energy. Since at equilibrium the entropy takes on a maximum value under the constraint that U is constant, it follows that the Helmoltz free energy attains a minimum at equilibrium for a system in contact with a heat reservoir that keeps a constant temperature in the system. In summary

-Ej/ kT - E j / KT Q( N , V , T )= e ; Pj = e / Q j A( N , V , T )= - kT ln Q ( N , V , T ) (24) dA= - SdT - pdV + m dN and 骣抖A 骣 ln Q S= -琪 = kln Q + kT 琪桫抖TN, V 桫 T N , V 骣抖A 骣 ln Q p= -琪 = kT 琪 (25) 桫抖VN, T 桫 V N , T 骣抖A 骣 ln Q m =琪 = -kT 琪桫抖NV, T 桫 N V , T

9 3. Grand canonical ensemble.

In the grand canonical ensemble the individual systems in the ensemble can exchange both energy and matter, hence the number of particles per individual system in the ensemble can change, while the volume is still the same for every element in the ensemble. Each element in the ensemble is an open system. The total energy and the total number of particles of the ensemble is conserved. This provides a Lagrange multiplier, b , associated with energy conservation, and a Lagrange multiplier,g associated with particle number conservation. This partition function is derived as the most likely distribution that preserves the total energy and the number of particles: i.e. maximize

å-Pj ln Pj - b(å Pj E j -U ) -g (å Pj N j - N ) - l(å Pj -1) j j j j -b E -g N . (26)  j j   ® ...® Pj = e e ; Z = å Pj , Pj = Pj / Z j U= P E; N = P N , P = 1 And we also find 邋 j j j j j from the stationarity condition with j j j respect to the Lagrange multipliers. The partition function for the grand canonical ensemble is given by

-bEj - g N j - b E j - g N j Z(b , g , V )= e e ; Pj = e e / Z (27) j The expression for entropy is hence given by

S= - k邋 Pjln P j = kb E j P j + k g 邋 N j P j + ln Z P j j j j j (28) =kb U + k g N + kln Z where as before, the sum over j runs over the accessible states in the ensemble. To provide the connection with thermodynamics:

10 1骣 S 1 =琪 = bk � b T桫 UN, V kT m骣 S m - =琪 =kg� g - T桫 NU, V kT (29) 骣抖E 骣 N p骣 S 1 j j -bEj - g N j =琪 =k[ -b琪 - g 琪 ] e e T桫抖 VU, N Zj 桫 V 桫 V

1 骣抖Ej 骣 N j 骣�( E jm N j ) = -邋[琪 -m 琪 ]Pj� p - 琪 P j Tj桫抖 V 桫 V j 桫 V Hence we obtain expressions for the Lagrange multipliers, but also a different expression for pressure for the grand canonical ensemble. Substituting the expressions for the Lagrange multipliers in the expression for S (Eqn. 28), we find S= U/ T -m N / T + k ln Z U- TS -m N = - kTln Z ( m , V , T ) (30) =U - TS - G = -( pV ) where the identification Nm = G is made (see further notes 2). Let me also note that this slightly unusual thermodynamic potential has been discussed in the notes “Fundamental Equilibrium Thermodynamics”. The natural variables for the grand canonical partition function are m , V, T, and these are also the natural variables of the characteristic function (pV). From pV= Nm + TS - U we readily derive d( pV ) = Ndm + m dN + TdS + SdT - TdS + pdV - m dN (31) =SdT + pdV + Ndm Summarizing

-Ej/ kTm N j / kT - E j / kT m N j / kT Z( T ,m , V )= e e ; Pj = e e / Z j (pV )= kT ln Z (m , V , T ) (32) d( pV ) = SdT + pdV + Ndm and therefore

11 骣抖(pV ) 骣 ln Z S=琪 = kln Z + kT 琪桫抖Tm,V 桫 T m , V 骣抖(pV ) 骣 ln Z p=琪 = kT 琪 (33) 桫抖Vm,T 桫 V m , T 骣抖(pV ) 骣 ln Z N=琪 = kT 琪抖桫mV, T 桫 m V , T Since entropy takes on a maximum at thermodynamic equilibrium under constraints of constant U and N, it follows that (pV ) also takes on a maximum, and can only increase in a spontaneous process in which the temperature and chemical potential remain constant through interactions with suitable reservoirs.

4. Isobaric-Isothermal ensemble.

In this ensemble the individual systems in the ensemble can exchange energy, and the volume in individual systems can adjust, while the particle number is constant in each system. The total energy of the ensemble, and the total volume of the ensemble is conserved. This provides Lagrange multiplier, b , associated with energy conservation, (and temperature). The second Lagrange multiplier, a , associated with the preservation of the total volume of the ensemble, is naturally associated with pressure, as this is equilibrated as the volumes adjust to equilibrium. This partition function can be derived by maximizing the most likely distribution under the constraints that total volume and total energy are conserved, i.e. maximize

å-Pj ln Pj - b(å Pj E j -U ) -a(å PjV j -V ) - l(å Pj -1) j j j j -bE -aV (34)  j j   ® ...® Pj = e e ; D(b,a, N ) = å Pj , Pj = Pj / D j The partition function for the isobaric-isothermal canonical ensemble is given by

-bEj - a V j - b E j - a V j D(N ,a , b ) = e e ; Pj = e e / D (35) j The expression for entropy is then given by

S= - k邋 Pjln P j = kb E j P j + k a V j P j + k ln D j j j (36) =kb U + k a V + k ln D

12 To provide the connection with thermodynamics: 1骣 S 1 =琪 = bk � b T桫 UN, V kT p骣 S p =琪 = ka� a T桫 VU, N kT (37) 骣抖E 骣 V m 骣 S 1 j j -bEj - a V j - =琪 =k[ -b琪 - a 琪 ] e e T桫禗 NU, V j 桫抖 N 桫 N

1 骣抖Ej 骣 V j 骣�( E j pV j ) = -邋[琪 + p 琪 ] Pj� m 琪 P j Tj桫抖 N 桫 N j 桫 N

Substituting the expressions for the Lagrange multipliers, we find S= U/ T + pV / T + k ln D (38) U+ pV - TS = - kTln D ( p , T , N ) G ( p , T , N ) Hence, the natural variables for the isobaric-isothermal partition function are N, p, T, and these are also the natural variables of the characteristic function, the Gibbs free energy G. A process is spontaneous under the conditions of this ensemble (i.e. p and T remain constant through interaction with a pressure and temperature reservoir) if the free energy decreases, again in direct relation to maximizing entropy under the constraints of preserving total energy and volume. In terms of the natural variables, the partition function is given by

-Ej/ kT - pV j / kT - E j / kT - pV j / kT D(N , p , T ) = e e ; Pj = e e / D j G( p , T , N )= - kT ln D ( p , T , N ) (39) dG= - SdT + Vdp + m dN and 骣抖G 骣 ln D S= -琪 = kln D + kT 琪桫抖TN, p 桫 T N , p 骣抖G 骣 ln D V=琪 = - kT 琪 (40) 抖桫pN, T 桫 p N , T 骣抖G 骣 ln D m =琪 = -kT 琪桫抖Np, T 桫 N p , T

13 5. The generalized ensemble.

Each system in this ensemble can have a variable number of particles, volume and energy. The ensemble is constrained such that the total number of particles, total volume and total energy is conserved. This provides three Lagrange multipliers, denoted a, b , g , which will be identified with the intensive variables pressure, temperature and chemical potential respectively. As before this partition function can be derived by maximizing the most likely distribution under the constraints that total volume, total number of particles and total energy are conserved, i.e. maximize

-å Pj ln Pj - b(å Pj E j -U ) -a(å PjV j -V ) - g (å Pj N j - N ) j j j j -bE -aV -g N (41)  j j j   ® ... ® Pj = e e e ; Y(a,b,g ) = å Pj , Pj = Pj / N j and we also find

U=邋 Pj E j, V = PV j j , N = P j N j j j j The partition function denoted Y is given by

-aVj - b E j - g N j - a V j - b E j - g N j Y(a , b , g )= e e e ; Pj = e e e / Y (42) j The expression for entropy is then given by

S= - k邋 Pjln P j =b k E j P j + a k 邋 V j P j + g k N j P j + ln Y j j j j (43) =bkU + a kV + g kN + kln Y To provide the connection with thermodynamics: 1骣 S 1 =琪 = bk � b T桫 UN, V kT p骣 S p =琪 = ka� a (44) T桫 VU, N kT m骣 S m - =琪 =kg� g - T桫 NU, V kT and, substituting the expressions for the Lagrange multipliers in the equation for S we find

14 S= U/ T + pV / T -m N / T + k ln Y U+ pV - TS -m N = U + pV - TS - G =0 = - kT ln Y Y =1 -kTln Y (m , T , p ) = U - TS + pV - N m = ... = 0 (45) From this we can derive -d( kT ln Y ) = dU - TdS - SdT + pdV + Vdp -m dN - Nd m (46) = -SdT + Vdp - Ndm = 0 using the usual expression for dU. This indicates that the thermodynamic variables p, T , m cannot be varied independently, and the above relation is precisely the Gibbs- Duhem relation. The results for the generalized ensemble are a bit different from the other ensembles, and the interpretation of the results is not entirely straightforward. Further discussion can be found in an advanced book, “Statistical mechanics” by Terell Hill, Chapter 3. Let us write the partition function

-Ej/ kT - pV j / kTm N j / kT - E j / kT - pV j / kT m N j / kT Y( T , p ,m )= e e e = 1; Pj = e e e (47) j This indicates that the generalized partition function is simply the normalization condition on the probabilities, and it apparently does not have to be calculated! Despite this seemingly attractive feature, or perhaps because of it, it is stated in the literature there is not much use for the generalized partition function (except that it provides the very useful Gibbs-Duhem equation).

6. Concise summary of all possible ensembles.

From the general ensemble we can deduce the general relation of the partition function to the Lagrange multipliers and then the thermodynamic quantities. In this section we will refer to each partition function as Y( T orU , porV ,m or N ) , depending on the variables

Ej, V j , N j that are held constant in the specific ensemble. We have the most general form of the probabilities

-bEj - a V j - g N j Pj = e e e/ Y (48) S= - k Pjln P j =b kU + a kV + g kN + k ln Y j

15 Making the identifications b=1/kT , a = p / kT , g = - m / kT this can be written as -kTln Y = U + pV - TS -m N = 0 (49) This formula can be taken to be more general in the sense that -TS is always present, U

is present if E j is variable, pV is present if the Vj are variable in the ensemble, while

-mN is present if N j is variable. This can be checked for the actual derivations we did for the various ensembles. Hence we can make the following 8 types of ensembles

Variable Partition Characteristic Name of Thermodynamic Differential quantities function function ensemble from statistical mechanics none kTln Y ( U , V , N ) TS Micro- dS= dU/ T + ( p / T ) dV - (m / T ) dN canonical

E j - kTln Y ( T , V , N ) U-TS=A canonical dA= - SdT - pdV + m dN

Vj kTln Y ( U , p , N ) TS-pV=U-G ? d( TS- pV ) = ... dU + ... dp + ... dN

N j kTln Y ( U , V ,m ) TS+ m N ? dH=... dU + ... dV + ... dm ! =U+pV=H (unusual variables !)

Ej, V j - kTln Y ( T , p , N ) U- TS + pV = G Isobaric- dG= - SdT + Vdp + m dN isothermal

Ej, N j kTln Y ( T , V ,m ) TS- U +m N = pV Grand- d( pV ) = - SdT + pdV + Ndm canonical

Vj, N j - kTln Y ( U , p ,m ) -TS - pV +m N = U ? dU= dU +0 dp + 0 dm !

Ej, V j , N j kTln Y ( T , p ,m ) TS- U - pV + m N General 0 = -SdT + Vdp - Ndm = 0 Gibbs Duhem

From the above table, it will be clear that not all results are equally obvious or familiar, and not all cases have their correspondence in commonly known functions of thermodynamics. In particular the enthalpy function looks a little suspect, as the natural variables appear to U, V and m . The interested reader may want to verify the results in this table, or find further literature on this. I drove the analysis to its logical conclusion, but did not pursue the physical interpretation very far.

7. Further notes.

16 1. In the literature one often finds expressions for the partition function in which the sums are performed not over individual states but over energy levels. To derive equations, it is much better to start from equations in terms of states. To show the dangers, let us try it the other way, using sums over levels. Let us consider the example of the canonical partition function and write

-bEj( N , V )1 - b E j ( N , V ) Q( N , V ,b )= W ( N , V , Ej ) e ; P ( E j ) = W ( N , V , E j ) e E j Q and evaluate the derivative of ln Q with respect to V. One might be tempted to write 抖E E 骣 lnQ 1 j-b E j j 琪 = -b W(N , V , Ej ) e = - b P ( E j ) = b p 抖邋桫 VT, N QEj V E j V

The end result is correct, but the mathematics is suspect as the number of states

W(N , V , E j ) (i.e. the partition function of the microcanonical ensemble) explicitly

depends on V and on Ej ( N , V ) , and its derivatives are not taken into account. Moreover, it assumes that all states of a given energy have the same derivative with respect to volume, which is not necessarily true, (certainly if other types of derivatives would be considered). Instead of the above, following simple rules of mathematics, I would have to evaluate

骣 ln Q 琪 = 桫 V T, N

1抖E-bE 1W( N , V , E ) - b E 1 禬 ( N , V , E ) E - b E -b W(N , V , E ) j ej + j e j + j j e j 邋 j 抖抖 QEj V Q E j V Q E j Ej V and using the derivatives of W as defined in the microcanonical ensemble (see subsection 1), one obtains

17 1抖E-bEp 1 - b E 1 1 E - b E = -b W(N , V , E )j ej + W ( N , V , E ) e j + W ( N , V , E ) j e j 邋 j抖 j j QEj V kT Q E j kT Q E j V = b p It follows that taking the derivative of W into account, the two terms indeed properly cancel, such that the proper result is obtained. It still means that the first way of deriving the result is essentially wrong, if no mention is made of this cancellation. Moreover, the above identification of the derivatives of W(N , V , E j ) for arbitrary energies is dubious, and it would probably be more appropriate to write for the last two terms p( E ) E 1j-bEj 1 1 j - b E j 邋 W(N , V , Ej ) e + W ( N , V , E j ) e = 0 QEj kT( Ej ) Q E j kT ( E j ) V such that cancellation occurs for each energy level E j separately. It is clear that the above derivation, using sums over energy levels is full of danger. There is an easier solution. In the development discussed in these notes, we always assume a sum over individual states, such that

Q( N , V ,b ) = e-b Ej ( N , V ) j

lnQ 1 抖E-b E E = -bjej = - b j P = b p 抖邋 j V Qj V E j V There are no ambiguities, and the mathematics is straightforward.

2. There is an interesting perspective to see that G= Nm . Since G( N , p , T ) and N are both extensive variables (they scale linearly with the size of the system), we can write G( N , p , T )= N g ( p , T ) , where g( p , T ) is independent of the size of the system. Hence

骣G m=琪 = g( p , T )� G ( N , p , T ) N m 桫N p, T or, the chemical potential is the Gibbs free energy per particle.