1 Entropy 1. 3 Extensivity 4. 5 Convexity 5

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1 Contents CONEX FUNCIONS AND HERMODYNAMIC POENIALS 1 Entroy 1 2 Energy Reresentation 2 3 Etensivity 4 4 Fundamental Equations 4 5 Conveity 5 6 Legendre transforms 6 7 Reservoirs and Legendre transforms Heat Reservoir hermodynamic otentials Gibbs equations hermodynamic Relations and hermodynamics Stability Mawell relations hermodynamic derivatives hermodynamic inequalities Physical consequences hermodynamic stability 15 1 Entroy here eists a function S, called the thermodynamic entroy, with the following roerties: (i) In a simle thermodynamic system, the entroy is a differentiable, concave, etensive function of E,, N; it is increasing in E and. (ii) he entroy of a comosite system is the sum of entroies of each simle subsystem. (iii) If two systems A and B, with initial total energy E tot = EA i + Ei B, are brought into contact and allowed to echange energy, then the final total equilibrium entroy will be S tot = su [S A (E A ) + S B (E B )] (1) E A,E B E A +E B =E tot his equation generalizes to more than two subsystems, and when more than one quantity is echanged. Definitions of, and µ (Gibbs equation) ds = 1 de + d µ dn. (2)

2 Partial Equilibrium When each subsystem of a comosite is in a state of internal equilibrium, but not in equilibrium with the other ones, we say that the comosite is in a artial (or local) equilibrium state. E A, A,N A E B, B,N B If Y is the variable (or several such variables) that secifies a artial equilibrium state of a comosite system, then the entroy of such a state is a function of Y and we shall denote it by S tot (Y ). he maimization of the entroy can be rewritten as Consider for simlicity only the energy, then S tot = su S tot (Y ). (3) Y S tot (E A ) = S A (E A ) + S B (E tot E A ), (4) where S A and S B are the equilibrium entroy functions of the two subsystems which are, each one, in a state of internal equilibrium. Maimization of S A (E A, A ) + S B (E B, B ) = S A (E A, A ) + S B (E tot E A, tot A ) yields 1 A 1 B = 0, A A B B = 0. Relation with the emirical notion of temerature Suose that the initial energy EA i is near the equilibrium value ĒA so that we may treat E A = ĒA EA i as a differential. hen S tot S A E A S ( B 1 E A E B E A = EB =EB i A i 1 ) B i E A 0 (5) EA =E i A hen E A > 0, if A i < B i, i.e., if the initial temerature of A is lower, there will be a heat flow from B to A as long as the temeratures do not equalize. Similarly, if A i > B i, then E A < 0, i.e., A will give energy to B, if its initial temerature is higher. his shows that the above definition of temerature is in agreement with the emirical notion of temerature. 2 Energy Reresentation Since S(E,, N) is invertible with resect to E, the inverse function E(S,, N) is differentiable, conve, etensive, increasing in S and decreasing in. hen the

3 maimization rincile of the entroy becomes: If two simle systems A and B, with initial total entroy S tot = SA i + Si B, are brought into contact and allowed to vary their entroies, then the final total equilibrium energy will be E eq tot = inf [E A (S A ) + E B (S B )] (6) S A,S B S A +S B =S tot his equation generalizes to more than two subsystems when volume and articles are echanged between the two system. Note While the rincile of maimum entroy is realized sontaneously in an isolated system when its comonents are initially not in equilibrium with each other, the energy minimization is not: the transformations that lead to the final state of equilibrium has to be slow enough, that is, quasi-static, so that the total entroy remains constant. Be that as it may, regardless of how the equilibrium state is reached, it is imortant to emhasize that it meets both conditions of maimum entroy and minimum energy. Entroy Reresentation iano Y αj

4 Energy Reresentation iano Y αj 3 Etensivity S(λE, λ, λn) = λs(e,, N), (7) for any λ > 0. Differentiating eq. (7) with resect to λ and taking it at λ = 1, one gets the Euler equations S = 1 E + µ N = or, equivalently, E = S + µn, (8) which can be read as an integrated version of Gibbs equation ds = 1 de + d µ dn 4 Fundamental Equations A functional relation between all etensive quantities of a thermodynamic system was called by Gibbs its fundamental equation. A fundamental equation determines comletely the thermodynamic roerties of the system. his is the case for S = S(E,, N) or E = E(S,, N). Each one of these two functions determines the temerature, the ressure, the chemical otential, and all the other thermodynamic coefficients of a system, such as its secific heats, comressibility, etc.. In brief, the entroy and the energy functions rovide then two equivalent reresentations of the fundamental equation of a system. he Gibbs equation in the entroy reresentation is ds = 1 de + d µ dn, (9)

5 whence the equations of state 1 = 1 S (E,, N) = E = S (E,, N) = µ = µ S (E,, N) = N,N E,N E, (10a) (10b) (10c) his means that,, N are not indeendent like the etensive variables E,, N, but are constrained by their derivative relations to S(E,, N), as given by eq. (10). he airs (1/, E), (/, ) and ( µ/, N) are usually called airs of conjugate variables in the entroy reresentation. Eamles Ideal fluid of van der Waals where s 0 is a constant. [ ( s = s 0 + k log (v b) e + a ) c ] v (11) Electromagnetic radiation S = 4 3 σ1/4 E 3/4 1/4 (12) 5 Conveity A subset D of a vector sace is conve is such that if for any and y in D also t + (1 t)y is in D for every t [0, 1]. Similarly, a function is conve if its grah curve is always uward, f (t + (1 t)y) tf() + (1 t)f(y). (13) Here, the domain D of f is assumed to be a conve set. A function is called strictly conve if the strict inequality holds.

6 y y = f() tf() + (1 t)f(y) f ( t + (1 t)y ) t + (1 t)y y A function g is called concave if g is conve. For a concave function tf() + (1 t)f(y) f (t + (1 t)y). (14) Note We need to allow functions to take the value +. Let D ess (f) be the essential domain of f, D ess (f) = { R n : f() < + }. One can show that any conve function is continuous in the interior of its essential domain. Definition he conve hull or conve enveloe of a set X of oints in the Euclidean lane or Euclidean sace is the smallest conve set that contains X. By conve hull of f we mean the conve hull of the grah of f. 6 Legendre transforms he L L of f : R n R { } is L(f) = f : R n R { } defined by L[f]() = f () = su [ f()] = inf [f() ] (15) y y = f() y = () Basic roerties (in any dimension):

7 (i) he Legendre transform of any function is conve. (ii) f = f whenever f is conve, i.e., on conve functions the oerator L is invertible and L 1 = L. If f is not conve, f is the conve hull of f. (iii) Linear ieces become cuss and conversely. (i) is easy to show: f (t + (1 t)q) = su [ (t + (1 t)q) f()] = su [t tf() + (1 t)q (1 t)f()] t su [ f()] + (1 t) su [q f()] tf () + (1 t)f (). (ii) secial case: f() is differentiable and is strictly conve, that is, conve with no linear arts. he L f () = su [ f()] is in general evaluated by finding the critical oint which maimizes the function F (, ) = f() hen, we can find the maimum of F (, ) using the common rules of calculus by solving F (, ) = 0 for a fied value of. Given the form of F (, ), this is equivalent to solving = f () for given k. Under the stated assumtions there eists a unique maimizer of F (, ). hus, we can write f () = f( ). where f ( ) = Eercise Show that under the stated assumtions f = f. Legendre duality f() sloe = () f () sloe = ()

8 f ( ) = f () = (16) o sum u, when f() is differentiable and is strictly conve we have f () = f( ), (17a) f() = f ( ). (17b) (iii): 9 Figure 3: Function having a nondifferentiable oint; its LF transform is affine. Note A function can be Legendre transformed only with resect to some variables, for eamle, f(, y) is transformed only with resect to y. In this case we shall write 173 Differentiable oints L of f : Each oint (, f ()) on the differentiable branches of y [f](q,...) = su [y q f(..., y)]. (18) 174 f () admits a strict suorting liney with sloe f () = k. From the results of 175 the revious section, we then know that these oints are transformed at the level Alternative 176 ofnotion f (k) into of Legendre oints (k, transform f (k)) admitting In the suorting alications, line of sloes it is useful f (k) = also. the following 177 form For eamle, of Legendre the differentiable transform branch of f () on the left (branch a in Figure 178 3) is transformed L[f]() into = ainf differentiable [ f()] branch = su of [f() f (k) (branch ]. a ) which etends (19) 179 over all k (, k l ]. his range of k-values arises because the sloes of the left- Since 180 branch of f () ranges from to k l. Similarly, the differentiable branch of f () 181 on the right (branch su [f() b) is transformed ] = into su [( ) the right differentiable ( f())], branch of f (k) 182 (branch b ), which etends from k h to +. (Note that, for the two differentiable we have 183 branches, the LF transform reduces to the Legendre transform.) L[f]() = L[ f]( ). (20) L 184has the Nondifferentiable same roertiesoint of Loflisted f : he above. nondifferentiable oint c admits not one but 185 infinitely many suorting lines with sloes in the range [k l, k h ]. As a result, each 186 oint of f (k) with k [k l, k h ] must admit a suorting line with constant sloe 7 Reservoirs 187 c (branch and c ). Legendre hat is, f (k) transforms must have a constant sloe f (k) = c in the 188 interval [k l, k h ]. We say in this case that f (k) is affine or linear over (k l, k h ). A heat reservoir (or heat bath) is a system (in thermal equilibrium) that is so large 189 (he affinity interval is always the oen version of the interval over which f that has to etract any amount of heat from it never affects its temerature A volume constant reservoir sloe.) is a system (in mechanical equilibrium) that is so large that to vary its volume never affects its ressure he case of functions having more than one nondifferentiable oint is treated simi- A article reservoir is a system (in chemical equilibrium) so large that to etract 192 articles larly by from considering it never each affects nondifferentiable its chemical oint otential searately. µ 0. he fundamental equation of a reservoir is given by Euler equation (8) Affine function S 0 (E 0, 0, N 0 ) = 1 E µ 0 N 0 (21) 194 Since f () in the revious eamle is conve, 0 f () 0 = f () 0 for all, and so the roles where 195 of f and f 0, 0 and can µ 0 be are inverted constants, to obtainwhile the following: the etensive a conve arameters function f () of having the reservoir an E 0, 0, and N 0 may vary when the reservoir interacts with other systems.

9 7.1 Heat Reservoir Suose now that a system is brought in thermal contact with only a heat reservoir over a eriod of time such it comes eventually to a final state of equilibrium. Since the system interacts only with a heat reservoir, it cannot vary its volume or its number of articles. Furthermore, suose that its initial temerature is the same as that of the reservoir 0, so that the transformation takes lace at constant temerature. he comosite system + reservoir is isolated, so that the total energy E tot is constant. he final equilibrium state corresonds to a maimum of the total artial equilibrium entroy given by eq. (4), which in this case reads S tot (E) = S(E,...) + S 0 (E tot E,...), (22) where S(E,...) is the entroy of the system; the dots stand for the other variables and N which are constants. Since 0 = and there is no variation of volume and article number, we can write eq. (21) as S 0 (E tot E,...) = E tot E + const = E where we have absorbed also the total energy in the constant, whence + const, (23) S tot (E) = S(E,...) E + const. (24) he final equilibrium state corresonds to the maimum of S tot [ S(E,...) E su [S tot (E)] = su E E ] + const, (25) which is attained at the value Ē = E(,...), solution of the etremum condition S E (Ē,...) = 1..., (26) that is, at the value of energy of the system, for which its temerature is equal to the temerature of the reservoir Crucially, the function in l.h.s of eq. (25) [ Ψ(,...) = su S(E,...) E ] Ē = S(Ē,...) E deends on and not on E because the equilibrium, maimized value of the total entroy is not anymore a function of the energy of the system, but only of its temerature, which is equal to that of the reservoir. Massieu function he function Ψ defined by eq. (27) is a Legendre transform of the system entroy S(E) in the variable 1/, ( ) Ψ(,...) = L 1 E [S],.... (28) In thermodynamic tetbooks, Ψ is a called Massieu function (with resect to the energy). From a hysical oint of view, it is nothing but the total entroy of a system and a heat reservoir in thermal contact, a art from an inessential constant term. (27)

10 Since S is concave, and the Legendre transform of any function is conve, Ψ is concave, because of the minus sign in the l.h.s of eq. (28). his is of course what was to be eected, given the hysical interretation of Ψ. Since the function Ψ is conve, the transformation eq. (28) is invertible: Ψ and S are related by the Legendre transforms [ Ψ (,...) = su E S(E,...) = inf >0 ] S(E,...) E [ ( ) 1 Ψ,... + E ] and the following roosition follows. he Massieu function Ψ(,...) rovides a reresentation of the fundamental equation of a system equivalent to the one given by the entroy S(E,...). (29) (30) Y ϑ Massieu function E E, E, N 1 1, 1, µ S E S E S E + µn 8 hermodynamic otentials hough Massieu functions lay a relevant role in statistical thermodynamics, in the develoment of classical thermodynamics, other equivalent functions have been introduced. hey are defined by S tot = inf Y [ ϑy S(Y,... )] Φ(ϑ,...). (31) for different choices of Y. he function Φ so defined is called the thermodynamic otential with resect to Y. he following roosition trivially follows from ro. above. he thermodynamic otentials rovide equivalent reresentations of the fundamental equation of a system. Useful rewriting utting in evidence energy Φ(, ˆϑ,...) = inf [E + ˆϑŶ ] S(E, Ŷ,... ). (32) Ŷ,E he minimization on E can be turned into a minimization on S, since inf [E + ˆϑŶ ] S(Ŷ,... ) = inf [E(S, Y,...) + ˆϑŶ ] S, Ŷ,E Ŷ,S whence Φ(, ˆϑ,...) = inf [E(S, Y,...) + ˆϑŶ ] S = L S,Y [E] (, ˆϑ,...). (33) Ŷ,S

11 From this formula we can read the conveity/concavity roerties of the otential: because of the minus sign in front of the Legendre transform, the otential is concave in the variables and ˆϑ, while is conve in the other etensive variables which are not affected by the Legendre transform (since the energy is conve in all its variables). his is summarized by the following roosition: he thermodynamic otentials are concave in their intensive variables and conve in their etensive variables. Most common otentials (a) Y = 0. he otential is the Helmholtz otential or free energy F (,...) = L S [E] (,...) = E(S,...) S (34) with S = S(,...) fied by the minimization. F is a concave function of and cove in variables and N. (b) Y =, ˆϑ =. he otential is the Gibbs otential or Gibbs free energy G(,...) = L S, [E] (,,...) = E(S,...) + S (35) with S = S(,...) and = (,...) fied by the minimization. G is concave in and and conve in N. (c) Y = N, ˆϑ = µ. he otential is the Landau otential or grand canonical otential Ω(, µ...) = L S,N [E] (, µ,...) = E(S,...) µ N S (36) with S = S( µ...) and N = N(, µ...) fied by the minimization. Ω is concave in µ and and conve in. (d) Y =, S =constant, ˆϑ =. he otential is the Enthaly H(S,...) = L [E] (, S,...) = E(S,...) + (37) with = (S,...) fied by the minimization. H is a concave in and conve in S and N. In the following, in agreement with the standard notations, we shall dro the bars and write F = E S, G = E + S, Ω = E µn S and H = E +. In case of ambiguity is however convenient to restore the bar notation. 8.1 Gibbs equations Consider the differential of the free energy F = E S But whence df = de ds Sd. ds = de + d µd, df = Sd d + µdn. (38) his is the Gibbs equation in the F -reresentation, from which one reads the equations of state F F = S, F,N =,,N N = µ. (39), Similar equations hold for the other otentials.

12 Second law Maimizing the total entroy of the comosite system + reservoir is equivalent to minimizing the otential. We can consider more general situations in which the maimization of the total entroy, given by eq. (3), can be turned into a minimization of the otential. he following roosition is nothing but a reformulation of the maimum rincile given by the second law. Let Y be an internal variable (or a set of variables) that secifies a state of artial equilibrium of a comosite in contact with a reservoir at constant ϑ. hen the equilibrium value Ȳ of Y is that for which the otential Φ = Φ(ϑ, Y ) has its minimum value Φ(ϑ,...) = inf Φ(ϑ, Y ) (40) Y Potential Gibbs equation Euler equation S(E,, N) ds = 1 de + d µ dn S = 1 E + µ N E(S,, N) de = ds d + µdn E = S + µn H(S,, N) dh = ds + d + µdn H = E + = S + µn F (,, N) df = Sd d + µdn F = E S = + µn G(,, N) dg = Sd + d + µdn G = F + = Nµ Ω(,, µ) dω = Sd d Ndµ Ω = F µn = Eamle: equilibrium of a water drolet in the fog he Landau otential is a natural function of the variables, µ, and, and is articularly convenient to use in the analysis of roblems involving constant and µ. Here is an eamle. Consider a liquid drolet in some form of continuous medium, such as a fog where there are water drolets in the air. he energy associated with the surface tension is a minimum when the drolet is a shere, and the drolets can therefore be considered as sherical. here must be a ressure difference between the inside and the outside of the drolets, because otherwise the tendency of the surface tension to contract the the drolet will not be balanced and t there would be no equilibrium. A urely mechanical calculation shows that the ressure difference required to balance the surface tension γ is = 2γ (41) R for a shere of radius R. his result was known to Lalace. We derive eq. (41) using thermodynamic arguments. he total system of interest is formed by the drolet (subscrit d), the eternal medium (subscrit e) and the surface (sherical) that searates them. One can not suose that the number of articles is constant because the articles may evaorate from the drolet in the medium, or condense from the medium on the drolet. In equilibrium, the chemical otential inside and outside are the same, and therefore it is convenient to choose a descrition of the medium in which it is fied µ. he only otential we have been introduced that involves µ as indeendent variable is the Landau otential Ω(,, µ). With g = 4πR 3 /3, A d = 4πR 2 for a shere, and γ a secific function of the temerature, we have the

13 correct variables to use Ω(,, µ) as thermodynamic otential, with A considered as a function of. From Euler s equation Ω = F µn = for a system in a single volume. In the resent case we have three contributions, Ω = d d e e + γa d due to the dro, the eternal medium and the surface of searation, resectively. In equilibrium, dω/dr must be equal to zero. We have d d dr = 4πR2 = d e dr (the sum of d and e remains constant when the radius of the dro changes) and da d /dr = 8πR. hus, we have dω dr = ( d e )4πR 2 + γ8πr = 0 he result eq. (41) follows with = d e. 9 hermodynamic Relations and hermodynamics Stability We shall now review the basic thermodynamic relations which can be easily obtained by means of the thermodynamic otentials and related them to the basic condition of thermodynamic stability. 9.1 Mawell relations he so-called Mawell relations are the equations that arise from the equality of the mied artial derivatives of the fundamental equation eressed in any of the various ossible reresentations. hey can be read directly from the table above. For eamle, from the differential of G (at N constant), one reads = S. (42) his is one of the Mawell relations, the others follow similarly. 9.2 hermodynamic derivatives he first derivative of the fundamental equations rovide the equations of state of the system and are given by either the intensive variables,, µ or by etensive variables S, E,, N, in the right combination and deendence on the other variables. For eamle in the G-reresentation we have S = S(, P, N), = (,, N), µ = µ = µ(,, N). he second derivatives of the fundamental equation describe the roerties of the materials which are of direct eerimental interest. hey are usually well known for many materials and tabulated, in articular those that are functions of and. Here are the basic ones (for constant N):

14 molar heat caacity at constant ressure coefficient of thermal eansion isothermal comressibility α = 1 c = S N = 2 G N 2 = 1 S = 1 2 G κ = 1 = 1 2 G 2 (43a) (43b) (43c) he last equality for each coefficient has been obtained from the Mawell relations in the G-reresentation. For systems of constant N, all other second derivatives can be eressed in terms of these three, in articular, the following ones, which are of ractical interest: molar heat caacity at constant volume c v = S = 2 F (44a) N N 2 adiabatic comressibility Using Mawell relations, it can be shown that c v = c α2 Nκ κ S = 1 = 1 S 2 H 2 S (44b) (45a) κ S = κ α2 Nc. (45b) Finally, we leave as an eercise to show that the molar heat caacities are also given by the equations c v = U, c = H, (46) which relate more directly to their measurements, and that eq. (45) imlies 9.3 hermodynamic inequalities c /c v = κ /κ S. (47) From the conveity roerties of the otentials, some hysically relevant inequalities follow. o see how this comes about, let us consider a system at constant N, so that there are only two indeendent variables. Concavity of entroy he second law establishes that the entroy is concave in all its natural variables. Such a global conditions entails local conditions for its second derivatives they must be non ositive and its Hessian it must be non negative that is, 2 S E 2 0, 2 S 2 0, E 2 S E 2 2 S 2 E ( 2 ) 2 S 0. (48) E Conveity of energy he energy is globally conve in in all its natural variables, therefore its second derivatives must be non negative and its Hessian must be non ositive 2 E S 2 0, 2 E 2 0, S 2 E S 2 2 E 2 S ( 2 ) 2 E 0. (49) S

15 Conveity/concavity of the Helmholtz otential With resect to the volume, the Helmholtz otential maintains the conveity roerty of the energy, while the minus sign in from of the Legendre transform indicates that it is a concave function of. hus, 2 F 2 F 2 0, 2 0, (50) Concavity of the Gibbs otential he Gibbs otential is globally concave. hus, 2 G 2 G 2 0, 2 G 2 0, 2 ( G 2 ) 2 G (51) Conveity/concavity of the enthaly With resect to the entroy, the enthaly is conve, but concave as a function of the ressure. hus, 2 H 2 H S 2 0, 2 0, (52) S 9.4 Physical consequences he most imortant hysical consequences of the thermodynamic inequalities are summarized by the following roosition: he addition of heat, either at constant ressure or constant volume, necessarily raises the temerature of a system more for a rocess at constant volume than for one at constant ressure. And the decrease in volume, whether isothermal or isentroic, necessarily increases its ressure more for an isothermal rocess than for an isentroic one. Let us rove this roosition. By comaring eq. (44) with eq. (50) and eq. (52), we infer that c v 0, κ S 0, (53) while the comarison of eq. (43) with eq. (51) gives From eq. (45a) and eq. (47), we finally obtain c 0, κ 0. (54) c c v 0 κ κ S 0, 10 hermodynamic stability If the entroy were not concave, equilibrium would not be stable. U,, N U,, N

16 o show this, let us consider two identical systems, each with fundamental equation S = S(E), in a container with rigid and adiabatic walls. In the middle of the container is laced a iston. By symmetry, ressures and temeratures to the right and left of the iston are equal, therefore the system is in equilibrium. We eect it to remain in this state as long as the constraints are not modified. However, if the deendence of the entroy on the energy were like this S(U + U) 1 [S(U + U)+S(U U)] 2 S(U) S(U U) U U U U + U this would not haen: there would be a transfer of energy E from one system to another because the entroy value corresonding to this echange, S(E + E) + S(E E), would be higher that the initial value 2S(E). So that the equilibrium states are stable is therefore necessary that S is a concave function of energy, i.e., such that, for any value of E and E, S(U + E) + S(U E) 2S(E). which is indeed the condition of concavity of eq. (14) (for t = 1/2, = E + E, y = E E).