legendre transforms and other potentials

7/28/2019 Legendre Transforms and Other Potentials

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Chapter 7

Legendre transforms and other potentials

7.1 New thermodynamic potentials from bathsWe began our discussion of equilibrium by considering the manner in which the entropy

emerges as important for isolated systems. A particularly telling fact was that the

independent variables of the entropy, ,,, are naturally constant in isolated systems. Inpractical settings, however, systems are rarely isolated, and it is often difficult to control

precisely the energy and volume of a system when outside forces are acting on it. Instead,

it is easier to control so-called field parameters like temperature and pressure.

Here, we will discuss the proper procedure for switching the independent variables

of the fundamental equation for other thermodynamic quantities. In doing so, we willconsider systems that are not isolated, but rather held at constant temperature, pressure,

and/or chemical potential through coupling to various kinds of baths. We will find that, in

such systems, entropy maximization requires us to consider the entropy of both the system

and its surroundings. Moreover, new thermodynamic quantities will naturally emerge in

this analysis: additional so-called thermodynamic potentials.

To achieve a system that is maintained at a constant temperature or pressure, one

couples the system to a bath, at which point it is no longer isolated. As we have discussed

before, a bath is essentially a large reservoir of energy, volume, or particles that can

exchange with the system of interest. The bath is so large that exchanging amounts ofvolume/energy/particles with the system that change the system state are in fact so small

for the bath that it is essentially always at the same state. Some examples:


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Recall our earlier result that systems that can exchange // reachthermal/mechanical/chemical equilibrium by finding the same //. Here, the maindifference with these previous results is that the bath is so much larger than the system

that it stays at the same //, and hence it sets the final equilibrium values of theseparameters. We consider the bath to be ideal in that it is infinitely large.

7.2 Constant temperature coupling to an energy bathLet us consider the specific case in which a system comes to thermal equilibrium with an

ideal energy bath, i.e., when the system is held at constant temperature. In reaching

equilibrium, the entropy of the world is maximized subject to energy conservation:

world = + bathmaximum at equilibriumworld = + bathconstant

(7.1)

Here, the unsubscripted variables are those pertaining to the system. Since the bath is solarge relative to the energy transfer between it and the system, its state does not change for

any measurable energy transfer with the system. We can therefore write the integrated

form of its fundamental equation with the approximation that all of its intensive state

parameters are constant:

bath = bathbath +bathbath

bath bathbath

bath

= bathbath + constants(7.2)

where bath is constant due to the constancy of state but bath depends on the exchangewith the system. We also know that the energy of the bath can be found from the total

energy and that of the system; therefore,

bath = world bath + constants(7.3)

Since the energy of the world does not change, we can absorb world into the constantsgroup:

bath

=

bath + constants

(7.4)

Finally, we combine this expression with the entropy of the system:

world = bath + constants(7.5)

To find the point of equilibrium between the two systems, we maximize this expression

with respect to the energies of the system and bath. Note here that bath is constant


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because we are dealing with a bath; therefore, it does not affect the maximization

procedure. Since bath is the final equilibrium temperature of our system, we will simplywrite it as . Moreover, the constants group is unaffected by the energy distribution, andhence we can neglect its presence in the equation above. On the other hand, is dependenton

, since changes to the energy of our system are generally large enough to affect the

value of the entropy function. As a result, we might write the equilibrium expression as:

maxworld max , , (7.6)That is, equilibrium occurs at the energy of the system that maximizes this expression,

where is treated as a constant. It turns out that, by convention, thermodynamics hasdefined a quantity that entails this particular combination of variables, albeit in a slightly

rearranged manner. The Helmholtz free energy is so defined:

(7.7)

We note that sometimes the Helmholtz free energy is written as in the physics literature,a mere matter of convention. Now we can write, for the equilibrium condition,max (7.8)

Equivalently, we can write (since is constant):min (7.9)

That is, the Helmholtz free energy is at a minimum at equilibrium.

is a new function that

we have introduced. This function is specific to the system; it does not rely on theproperties of the bath, and it is therefore a new thermodynamic potential of the system.

What is interesting to us is that the equilibrium value of is not a function of,,, as inthe fundamental equation for the entropy, but it is a function of ,,. That is, theindependent variable has been replaced by in the case in which our system is held atconstant temperature.

How do we know that this change of variables has occurred? First, we know thatis determined by the bath. It is something we specify, since we choose the particular kind

of bath that we bring in contact with the system. is therefore an independent variable.Second, the value of

of our system is determined by the entropy maximization /

Helmholtz free energy minimization procedure. We cannot set independently atequilibrium, since its value is found by entropy maximization via coupling to the bath.At equilibrium for closed systems held at a constant temperature and

volume, the Helmholtz free energy is at a minimum. hasindependent variables ,,, so that we have the function(, , .

EXAMPLE 7.1


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Find the Helmholtz free energy of a monatomic ideal gas, whose fundamental entropy function can be

written as

(,, = ln ( !( + "where

"is a constant.

Conceptually we will put the gas in equilibrium with a temperature bath with which it can exchange

energy. The Helmholtz free energy is minimized, and its form is, = = ln ( !( +"

We minimize with respect to for fixed ,,,0 = $%%& = 1 32

Solving for

,

= 32 Back-substitution into the Helmholtz free energy expression finally gives,

= 32 ln (3 2 !( +"= *32 ln (3 2 !( +"

which is only a function of,,, as we should have.7.3 Complete thermodynamic information and natural variablesSince we have just introduced a new function (, , , it is worthwhile for us to discussthe completeness of the thermodynamic information contained in it. Recall that only three

pieces of information are required in order to completely specify the (extensive)

thermodynamic state of a single-component system. Once these are specified, all other

thermodynamic variables have a unique value. The complete list of variables includes,,,,, ,. If we pick three of these, then we require a thermodynamic function thatcan provide the values of the other four. In the case of the entropy, the three variables to

be specified are ,,; then the function (,, gives the entropy directly, and ,,extend from its partial derivatives. In that sense, (,, is a function that gives acomplete set of thermodynamic information.

When we swapped for in our constant temperature example, we had to switchfrom to as the relevant potential for our system. One might wonder why we were notable to use the function (,, instead of (, , , where we have done the


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appropriate mathematical manipulations to convert(,, into (,,. It turns outthat such an approach would result in a loss of thermodynamic information. That is, we

would not be able to manipulate (,, so as to extract the complete set ofthermodynamic variables at a given state point. For example, what if we wanted to

compute

from

(,,? We would take the following approach:

= 1 + $&-,. = 1 $&-,.

= / $&-,. + (, (7.10)

In our extraction of, we end up with an unknown and -dependent constant due to thefact that we needed to perform an integration. We can never find the value of that constant

from just the function (,,. We would need some other information. In contrast, ifwe knew (,,, we would never find any unknown constants in the determination ofthe other thermodynamic properties (,,) because all of these come from derivatives ofthe function of interest. In this sense, we call (,,fundamental, in that it contains allthermodynamic information, while (,, is not. We also say that the naturalvariables of the thermodynamic potential are ,,. These are general terms inthermodynamics that apply to all potentials.

thermodynamic potential natural variables fundamental form

entropy

,,

(,,

internal energy ,, (,,Helmholtz free energy ,, (,,

As alluded to above, the Helmholtz free energy function (, , is also fundamental.That means that we dont lose any thermodynamic information in (, , , despite thefact that it represents a function where we now have as an independent variable ratherthan . This means that all other thermodynamic properties extend from derivatives of.This can easily be shown by computing the differential form for:

= ( = = (+ (7.11)where we have substituted the differential form for the fundamental energy equation. A

little simplification shows that,

= + (7.12)


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This expression means that the partial derivatives of the Helmholtz free energy are given

by:

$

&

-,.

= $&1,.

= $&1,-

= (7.13)We see that a complete set of thermodynamic information can be derived from, given thecurrent state conditions specified by ,,. Therefore, we conclude that(, , is thefundamental thermodynamic potential at constant,, conditions.7.4 Legendre transforms: mathematical conventionAbove we derived by considering the physical process in which a system is attached to aheat bath. Mathematically speaking, however, there is a deep theoretical foundation for

transforming the energy version of the fundamental equation,

(,,, into the

Helmholtz free energy(, , . Per the considerations above, we can find the value ofat any state point using the following mathematical construction:(,, = (, , (7.14)

where is the value of the internal energy that minimizes the expression on the righthand side, for given values of,,. Equivalently, we can switch to the energy version ofthe fundamental equation to show that

(,, = (, , (7.15)where now is the value of the entropy that minimizes . In either case, we find the valueof one independent variable that minimizes the Helmholtz free energy.

These constructions are closely related to a mathematical operation called the

Legendre transform. Consider a generic function (. Let the derivative of that functionbe 4=. A Legendre transform produces from( a new function 6(4, where theindependent variable has changed to 4. This new function contains all of the sameinformation as the original function, so that it can easily be transformed back:

( = 6(4= max

84

= 4 where solves = 4(7.16)

In other words, the Legendre transform produces a new function equivalent to 4 ,where 4 is independently fixed and is the value of that maximizes this function forgiven 4.


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To show that the Legendre transform maintains all of the information as the original

function, we merely need to show that6(4 can be converted back to(. The inverse of aLegendre transform turns out to be itself. We compute the derivative of 6(4 todemonstrate this:

6(44 = 4 4 (= 4 (44 + ( (44 = 4 (44 + 4 (44 9since ( = 4:=

(7.17)

Then, computing (


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6(44 =


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This set of equations is exactly a Legendre transform. In fact, we can write the expression

for the Helmholtz free energy as the entropy Legendre transform of the internal energy:

(,, = F( ,,

min at equilibrium (7.25)

Here, the subscript indicates a transform with respect to the independent variable .In performing a Legendre transform of the internal energy, we swapped for as

an independent variable to produce the Helmholtz free energy function. But this is notthe only Legendre transform that we can perform, since there are three independent

variables. We can consider Legendre transformations between any two conjugate

variables. This means that we can also swap for and for , since Legendretransforms trade an independent variable for the derivative of the thermodynamic

potential with respect to it. We cannot, however, swap non-conjugate pairs, like for . Ingeneral, Legendre transforms are the mathematical equivalent of connecting our system to

various kinds of baths:Legendre transforms enable one to switch an independent variable of a

thermodynamic potential to its conjugate. The function produced

corresponds to the relevant thermodynamic potential at constant conditions

of the new independent variables, i.e., when the system is connected baths

corresponding to the Legendre-transformed variables.

7.6 The Gibbs free energyLets examine this principle in a slightly different context, in the case where a systemreaches equilibrium at constant and . In other words, the system is in contact withbaths that can exchange both energy and volume. Let us first proceed using the second law

as we did before,

world = + bathmaximum at equilibriumworld = + bathconstantworld = + bathconstant

(7.26)

were the very large bath admits the behavior,

bath = bathbath +bathbathbath + constants

(7.27)

Using this expression and expressing the bath quantities in terms of the difference between

the world and system gives,


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world = bath bathbath + (constants (7.28)The maximum entropy equilibrium condition therefore becomes,

maxworld m a x (7.29)where we have replaced bath with and bath with since it is through the bath that thesystem pressure and temperature are ultimately maintained. Another way of representing

the right hand side is through multiplication by ,min+ (7.30)

Given our earlier results, this is equivalent to

min+ (7.31)

The quantity in brackets is called the Gibbs free energy, I + . Here,equilibrium is obtained by minimizing this expression with respect to the values of and, for given parameters and .

We could have also tackled this problem using Legendre transforms. Since the

system is connected to a bath with which it can exchange energy and volume, this means

that we swap both and by performing two Legendre transforms. We can startwith the Helmholtz free energy, where we have already performed a Legendre transform in

the entropy. Now we perform an additional Legendre transform in the volume. The

resulting potential is the Gibbs free energy:

I, , = -(,,I + min at equilibrium (7.32)Notice that the Gibbs free energy is naturally a function of the variables , , . Moreover,like the Helmholtz free energy, it is a minimum at equilibrium. We can examine the partial

derivatives ofI in the following manner: I = +

= + + (7.33)

Where we have substituted the differential expression for . Simplifying,I = + + (7.34)

Our differential expression for the Gibbs free energy simply states that the partial

derivatives are given by:


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$I&J,. = $I&J,. = $I&1,J = (7.35)Note that we can also relate the Gibbs free energy directly to the internal energy:

I(, , = -F( ,,>

I = + (7.36)In either case, we would have arrived at the same expression for the Gibbs free energy if we

had connected our system to energy and volume baths, and proceeded to maximize the

entropy of the world, similar to our original approach using the Helmholtz free energy. The

Legendre transform provides the mathematical equivalent of this procedure.

7.7 Physical rationale for Legendre transformsWhy do Legendre transforms naturally emerge when we consider alternate constant

conditions? The reason stems from the fundamental form of the bath entropy. In general,

we can express the entropy of a bath in terms of its extensive properties KL,bath, K!,bath, ,and the intensive conjugates of those bath = P


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maxworld maxST (KL, K!,


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Consider as an example a system connected to a thermal bath,

maxworld maxV WmaxX (,,,U Y (7.44)Since

is constant, we can rewrite this as,

maxworld minV WminX (,,,UY=minV (,,,U (7.45)Thus, the Helmholtz free energy is minimized at constant ,, with respect to anyunfixed internal degrees of freedom. Similarly for the Gibbs free energy,

maxworld maxV WmaxX,- (,,,U Y=minV

WminX,-

(,,,U + Y=minV I(, ,,U

(7.46)

The general idea of these results is that,

The relevant potential at a set of constant conditions is extremized not only

with respect to the variables describing what is being exchanged with baths,

but also with respect to any unfixed internal degrees of freedom.

7.9 The enthalpy and other potentialsNow we will shift our perspective slightly. Historically, much of classical

thermodynamics is built on Legendre transforms of the energy rather than the entropy,

although the formulations are ultimately equivalent. How can we modify the physical bath

scenarios for this particular situation? The fundamental equation can be written as(,,. As we described in chapter 4, conceptually we can hold a system at constantby always adjusting its energy so that the entropy stays at some fixed value. If we maintain

constant conditions in this way, but allow our system to exchange other quantities likevolume or mass, then

world

= + bath

= bath + (constants(7.47)

This simplification stems from the fact that energy is always exchanged in such a way as to

maintain constant entropy conditions for the system. Lets now assume that the system

exchanges volume with the bath as well, in order to maintain constant pressure conditions.

Continuing as we have before by expanding the bath entropy,


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world = bathbath + bathbathbath + (constants= world bath + bath(world bath + (constants= + (

constants

bath (7.48)

Since the temperature of the bath is constant, maximization of the entropy of the world can

finally be written as,

maxworld min- (,,+=min- Z (7.49)In the last line we have introduced the enthalpy. This important thermodynamic quantity

is essentially a Legendre transform of the fundamental energy equation,

Z(,, = -(,,Z + (min at equilibrium (7.50)The enthalpy is the thermodynamic potential corresponding to constant,, conditions,and it is a minimum at equilibrium. Like the internal energy, it relates to constant entropy

conditions, but at constant pressure rather than volume. The differential form of Z isfound using the same approach as before, the final result being:

Z = + + (7.51)which summarizes the partial derivatives

$Z &J,. = $Z&F,. = $Z&F,J = (7.52)The enthalpy is not called a free energy for the reason that the temperature is not

constant. The term free comes from the fact that and I have the term subtractedfrom the internal energy , which represents the part of the energy due to unorganizedmolecular heat motion. By subtracting this off, the remaining parts of the and Ipotentials represent energy potential available or free to do useful work.

The enthalpy may seem like an unusual potential that is not particularly useful due

to the fact that systems rarely achieve constant,, conditions. However, this potentialnaturally emerges as a useful quantity in a number of scenarios, such as isobaric heating,steady-state flow, and phase transitions. Even though the constant conditions are not, , in many of these case, it can be shown that the enthalpy nonetheless contains therelevant information.

What about other thermodynamic potentials? We have only pursued various

Legendre transforms of the entropy and volume, but we can Legendre-transform in the


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particle number as well. If there are multiple components, we will be able to transform

each separately, so that the number of possible new thermodynamic potentials quicklybecomes quite large. The resulting potentials are just as equally as valid as those we have

discussed so far; however, it turns out that their utility in common practice often is small,

simply because systems do not usually exist at constant chemical potential conditions for

all components.

7.10 Integrated and derivative relationsLets examine the integrated relationships for, I, and Z. Recall that the internal energy isgiven in integrate form by = + . We were able to show this using Eulerstheorem for extensive quantities. We can now use this expression to simplify the

integrated relation for the other potentials,

= = + I = + = Z = + = + (7.53)

We could have also arrived at these expressions by applying Eulers theorem directly to the

new thermodynamic potentials themselves, using the extensivity relations

(,[,[ = [(,,I, ,[ = [I(, ,Z([,,[ =[Z(,,

(7.54)

Here, only the extensive variables are scaled in each relation.

Notice in particular that the chemical potential in single component systems is

simply the per-particle Gibbs free energy, = I . In general, for multicomponentsystems, we have

= + P QQ = + P QQ

I = P QQZ = + P QQ(7.55)

These are the integrated relations describing the four fundamental thermodynamic

potentials. There are also ways to relate these potentials using derivatives. Consider, for

example, the relationship between and:


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= + = $&-,. (7.56)

Dividing by

!,

! = ! 1 $&-,. = $&-,. (7.57)

We could have also performed the transformation in the opposite direction, by expressing as an -derivative of. Moreover, we can use the expressions I = + , I = Z ,and Z = + to generate more such relations. In all, we find that

! = $&-,. ! = $Z&F,.

! = $&-,. ! = $I&,. I! = $Z &J,. I! = $&,. Z! = $I&J,. Z! = $&F,.

(7.58)

And of course, there are other such relations for the many other thermodynamic potentials

that can be constructed by Legendre-transforming the particle number variables inmulticomponent sytems. In general, it is not advised to attempt to memorize these; rather,

it is important to understand the origins of these relations such that they can be re-derived

at any time.

7.11 Multicomponent and intensive versionsThe most general forms for each of the thermodynamic potentials we have described

involve multiple particle number variables to indicate the amounts of different species. Let

us denote the collection of these variables as \Q] = \L, !, ]. The table belowsummarizes these potentials and their interrelationships.potential differential form integrated form

(,,\Q] = 1 + P Q QQ = + P QQQ


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(,,\Q] = + P QQQ = + P QQQ Z(,,\Q] Z = + + P QQQ

Z = + = + P QQQ

(, , \Q] = + P QQQ = = + P QQQ I(,,\Q] I = + + P QQQ I = + = + = Z

= P Q

Q

Q

An equivalent way of writing each of the sums involving chemical potentials would be to

use vector notation, ^ = ( L, !, and _ = (L, !, . Then we can take advantage ofvector dot products as, QQQ = _ ^ and QQQ = _ ^. This alternate notation isfrequently encountered.

When the system contains only a single component, we can define per-particle or

molar versions of each of the thermodynamic potentials. The procedure is identical to

what we did earlier for the energy fundamental equation, in Chapter 5, and can be applied

to both the differential and integrated forms of the potentials. As before, we will use lowercase letters to indicate quantities that are normalized by . The table below summarizesthe intensive potentials for a single-component system.

per-particle

potential

differential form integrated form and

relations

(b,c d = 1 ? + e = d + ? + e b(,c ? = d e = ? d + ef(,g = d + e = ? + e = dh(,c j=de j = ? d = j + e


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k(,g 6=d+e 6 = ? + e d= j + e= d = 6For multicomponent systems, converting the potentials to intensive form requires more

consideration.

7.12 Summary and look-aheadLets review the main points of our discussion. If we want to consider a system at

conditions other than constant ,,, we use a different thermodynamic potential otherthan

(,,or

(,,. That potential can be constructed by hooking our system to a

bath (or baths) of some kind. To arrive at the correct thermodynamic potential, we canalso perform a Legendre transformation, which generally constructs a new function similar

to 6 = ( . Thermodynamic potentials can then be related in both integratedand derivative forms.

Before we close this topic, we will hit on one point that we will revisit later during

our discussion of statistical mechanical ensembles. Earlier we discussed constant conditions as achieved by the point where we coupled our system to a heat bath and foundmax( + surr. Remember, however, that this maximization is really an approximationused to find the point at which there is a maximum number of microstates. This

approximation works well typically because the distribution of microstates is so sharplypeaked around one value of the energy transfer between the system and its surroundings.

However, despite the utility of this maximum term approximation, there can still be

fluctuations around the maximum as slightly different amounts of energy are exchanged

between the bath and the system. Indeed, these fluctuations are small and rare, since they

correspond to states of the system with greatly diminished numbers of microstates, but

they nonetheless exist, rigorously. That is, even though we cannot detect small variations

in the amount of energy exchanged between the system and the bath, these variations do

exist such that the system energy is never rigorously constant.

This turns out to lead to an important generality: only one of each conjugate pair of

variables can be truly constant at a time. If the temperature is held constant, by using an

infinite heat bath, then the energy of the system experiences slight fluctuations. If the

volume is held constant, then the pressure fluctuates from time to time. This point is not

really relevant at the macroscopic level, since the fluctuations are so small, but it is an

essential feature at the microscopic level.

EXAMPLE 7.3


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An experimentalist determines the heat capacity function of a substance to obey the empirical relation"-(, = !!, where is a constant. The experimentalist also finds the entropy and energy to bezero at absolute zero, for all volumes. Compute the thermodynamic potentials(, , Z(, , (, , andI(,. Assume constant conditions throughout. (The parameter will have a 1/N dependence so that the heat capacity remains extensive.)

To solve this problem, we start with the definition of heat capacity,

"- = $&-Using the fundamental equation at constant volume and particle number, = ,

"- = $&-So, we can perform integrals to compute the energy and entropy as a function of temperature:

= / "- = 3

!

+const

(

= / "- = l2!! + constBut since we know the entropy and energy are both zero at = 0, the constants are zero:

= l3! = l2

!!To find , we need to solve for the temperature as a function of entropy:

= $ 2

!&

L!

= 3 $ 2!&! ! = $ &L! !qLTo find the Helmholtz free energy, we use(, =

= l3! Wl2

!!Y= l6

!For the enthalpy, we need to find the pressure. One way is to use the energy derivative:

= $&F = $ 9l&L! !q!

Another way is to take the Helmholtz free energy derivative:

= $&1 = l3


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It is easy to verify that the two pressure expressions are equal, using = (2!!. Since theenthalpy depends on the entropy, and not the temperature, we use the former. First we express

volume in terms of pressure, since pressure is the independent variable in the enthalpy:

= $ &Ls

qL

!s

Then, constructing the enthalpy and substituting in for any terms,Z = + = $ &

L! ! t$ &Ls qL!su

qL+ t$ &

Ls qL!su= $12 &

Ls L!sFinally, we can find the Gibbs free energy. First, we express the volume as a function of

temperature and pressure, based on the pressure expression derived from the Helmholtz energy: = $3& qConstructing the Gibbs free energy from the Helmholtz free energy:I = +

= $3& q! + $3& q= $ 32& !q

We could have also formed

Iusing

I = Z . The result would have been the same.

legendre transforms and other potentials

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