Convex of convex functions with applications to Statistical Mechanics

Constantin Z˘alinescu University Alexandru Ioan Cuza Ia¸si Faculty of Mathematics [email protected]

Melbourne, MODU2016

C. Z˘alinescu Series of convex functions and entropy minimization Motivation

In Statistical mechanics or/and Statistical physics one considers the following problem: Having an isolated system with energy E, and total number of particles N, one distributes N particles over the energy levels (ej )j∈I . The total number of ways (of doing this) is N! Ω = Q . j nj ! The problem is to maximize Ω by keeping fixed E, N, that is X X nj = N, nj ej = E. j j

C. Z˘alinescu Series of convex functions and entropy minimization The usual procedure to solve this problem (found, practically in all books on Statistical mechanics and Statistical physics) is: By Stirling’s approximation, for large N, we have ln N! ≈ N ln N − N. Hence P P ln Ω ≈ N ln N − N − j (nj ln nj − nj ) = N ln N − j nj ln nj . Taking the Lagrangian     X X X L = N ln N − nj ln nj + α  nj − N + β  nj ej − E , j j j

∂L α+βe one finds = 0 for every j, and so nj = e j with ∂nj P α+βej P α+βej j e = N, j ej e = E. From these relations one finds (uniquely) α and β (which have physical interpretations). Moreover, ln Ω ≈ N ln N − αN − βE is the maximum entropy of the system.

C. Z˘alinescu Series of convex functions and entropy minimization As examples of (ej )j are: 1 2 2 2 ej = γ(j + 2 ), ej = γj(j + 1) (j ∈ N), eijk = γ(i + j + k ), (i, j, k ∈ N). α+βe Note that all nj (= e j ) are (strictly) positive and probably very few (maybe none) are integer. However, from the formulation of the starting problem we must have nj ∈ N, and so at most N of them might be non-zero.

C. Z˘alinescu Series of convex functions and entropy minimization The approach above can be found, for example, in: L. D. Landau, E. M. Lifshitz: Statistical Physics, Third Revised and Enlarged Edition, Pergamon Press Ltd., 1980 (see pp. 119, 120) T. Gu´enault,Statistical Physics, Reprinted revised and enlarged second edition, Springer, 2007 (see pp. 15, 16) R. K. Pathria, P. D. Beale, Statistical Mechanics, 3rd edition, Elsevier Ltd. (2011). J.-L. Basdevant, Les principes variationnels en physique, Vuibert, 2014 Alfred Huan: www.spms.ntu.edu.sg/PAP/courseware/statmech.pdf,

C. Z˘alinescu Series of convex functions and entropy minimization • Is it possible to prove rigorously that the solution found as above is indeed a solution for the problem P P P min n≥1 un(ln un − 1) s.t. n≥1 un = u, n≥1 σnun = v ? 2 • Which are the pairs (u, v) ∈ R for which the problem above has optimal solutions? • If the problem has not optimal solutions, which is the value of this problem?

C. Z˘alinescu Series of convex functions and entropy minimization Aim

Our main aim is to answer the questions above, at least partially. For this we prove the following result: Theorem

Let (E, τ) be a separated locally convex space and f , fn ∈ Λ(X ). P Assume that f (x) = n≥1 fn(x) for every x ∈ E. If ∗ x ∈ ∩n≥1 dom fn and (xn)n≥1 ⊂ (∂fn(x))n≥1 is such that ∗ P ∗ ∗ ∗ ∗ w - n≥1 xn = x ∈ E , then x ∈ dom f , x ∈ ∂f (x) and ∗ ∗ P ∗ ∗ f (x ) = n≥1 fn (xn). In particular,   ∗ ∗ X ∗ ∗ ∗ ∗ ∗ ∗ X ∗ f (x ) = min fn (xn ) | (xn )n≥1 ⊂ (dom fn )n≥1, x = w - xn n≥1 n≥1

Moreover, if f and fn are continuous on int(dom f ), and ∗ ∗ ∗ P ∗ x ∈ ∂f (x) with x ∈ int(dom f ), then x = w - n≥1 xn for some ∗ sequence (xn )n≥1 ⊂ (∂fn(x))n≥1.

C. Z˘alinescu Series of convex functions and entropy minimization Based on this result we study the problems

P ui P P minimize pi W ( ) s.t. ui = u, ui εi = v, i pi i i where pi ∈ [1, ∞) and W is one of the functions EBE , EFD , EMB below:  u ln u − (1 + u) ln(1 + u) if u ∈ R+, EBE (u) := ∗ ∞ if u ∈ R−,

 u ln u + (1 − u) ln(1 − u) if u ∈ [0, 1], EFD (u) := ∞ if u ∈ R \ [0, 1],  u ln u − u if u ∈ R+, EMB (u) := ∗ ∞ if u ∈ R−, ∗ with 0 ln 0 := 0 and R+ := [0, ∞[, R+ := ]0, ∞[, R− := −R+, ∗ ∗ R− := −R+; EBE , EFD , EMB are Bose–Einstein, Fermi–Dirac and Maxwell–Boltzmann (or Shannon) entropies, respectively.

C. Z˘alinescu Series of convex functions and entropy minimization Notation

–(E, τ) is a (real) separated locally convex space – E ∗ is the topological dual of E endowed with the weak∗ topology – Λ(E) is the set of proper convex functions from E into R := R ∪ {−∞, ∞} – Γ(E) is the set of those f ∈ Λ(E) which are lsc For f : E → R: – the domain of f is dom f := {x ∈ E | f (x) < ∞} ∗ – the conjugate of f is f : E → R defined by f ∗(x∗) := sup{hx, x∗i − f (x) | x ∈ E} – the subdifferential of f is ∂f (x) := {x∗ ∈ E ∗ | hx0 − x, x∗i ≤ f (x0) − f (x) ∀x ∈ E} if f (x) ∈ R, ∂f (x) := ∅ if f (x) ∈/ R – the directional of f at x with f (x) ∈ R is defined by 0 −1 f+(x, u) := limt→0+ t [f (x + tu) − f (x)] for u ∈ E; if f ∈ Λ(E) 0 then f+(x, u) exists in R for all x ∈ dom f and u ∈ E – For (Ai )i∈I a family of (nonempty sets), (ai )i∈I ⊂ (Ai )i∈I means that ai ∈ Ai for all i ∈ I C. Z˘alinescu Series of convex functions and entropy minimization Definition (Zheng, 1998)

Let A, An ∈ P0(E) := {F ⊂ E | F 6= ∅} (n ≥ 1). One says that P (An)n≥1 converges normally to A (wrt τ), written A = τ- n≥1 An, if: P (I) for every sequence (xn)n≥1 ⊂ (An)n≥1, the series n≥1 xn τ-converges and its sum x belongs to A; τ (II) for each (τ-)neighborhood U of 0 in E (that is U ∈ NE ) there P is n0 ≥ 1 such that τ- k≥n xk ∈ U for all sequences (xn)n≥1 ⊂ (An)n≥1 and all n ≥ n0 (observe that the series P k≥n xk is τ-convergent by (I)); (III) for each x ∈ A there exists (xn)n≥1 ⊂ (An)n≥1 such that P x = τ- n≥1 xn.

Observe that A in the above definition is unique; moreover, A is convex if all An are convex.

C. Z˘alinescu Series of convex functions and entropy minimization Subdifferential of a countable sum

The next result is useful for deriving the formula for the subdifferential of the sum. Theorem 1 (formula for the directional derivative of f ) P Let f , fn ∈ Λ(E) be such that f (x) = n≥1 fn(x) for every x ∈ E. Assume that x ∈ core(dom f ). Then

0 X 0 f+(x, u) = fn+(x, u) ∀u ∈ E. n≥1

C. Z˘alinescu Series of convex functions and entropy minimization Theorem 2 (uniform convergence) P Let f , fn ∈ Λ(E) be such that f (x) = n≥1 fn(x) for every x ∈ E. P Assume that the series n≥1 fn converges uniformly on a neighborhood of x0 ∈ int(dom f ). Then for every x ∈ int(dom f ) τ there exists a neighborhood U ∈ NE with x + U ⊂ dom f such P 0 0 that the series n≥1 fn+(·, ·) converges uniformly [to f+(·, ·)] on (x + U) × U.

C. Z˘alinescu Series of convex functions and entropy minimization Theorem 3 (formula for the subdifferential of f ) P Let f , fn ∈ Λ(E). Assume that f (x) = n≥1 fn(x) for every x ∈ E. If f and fn are continuous on int(dom f ), then

∗ X ∂f (x) = w - ∂fn(x) ∀x ∈ int(dom f ). n≥1

C. Z˘alinescu Series of convex functions and entropy minimization When E is a normed vector space, one has also the next result. Theorem 4

Let E be a normed vector space and f , fn ∈ Λ(E). Assume that P f (x) = n≥1 fn(x) for every x ∈ E. If f and fn are continuous on P int(dom f ) and the series n≥1 fn converges uniformly on a nonempty open subset of dom f , then X ∂f (x) = k·k - ∂fn(x) ∀x ∈ int(dom f ); n≥1 P moreover, limn→∞ k≥n ∂fk (x) = 0 uniformly on some neighborhood of x for every x ∈ int(dom f ), where kAk := sup {kx∗k | x∗ ∈ A} for ∅= 6 A ⊂ X ∗.

C. Z˘alinescu Series of convex functions and entropy minimization Corollary 5 P Let f , fn ∈ Λ(E). Assume that f (x) = n≥1 fn(x) for every x ∈ E, and f , fn are continuous on int(dom f ) for every n ≥ 1. Take x ∈ int(dom f ). Then

(i) f is Gˆateauxdifferentiable at x if and only if fn is Gˆateaux differentiable at x for every n ≥ 1. (ii) Moreover, assume that E is a normed vector space. If f is Fr´echetdifferentiable at x then fn is Fr´echetdifferentiable at x for every n ≥ 1.

C. Z˘alinescu Series of convex functions and entropy minimization Proposition 6 (to be continued)

σnx P Let fn(x) := pne with pn ≥ 1 for n ≥ 1, x ∈ R; f = n≥1 fn. (i) If x ∈ dom f then σnx → −∞, and so either x > 0 and σn → −∞, or x < 0 and σn → ∞. Furthermore, assume that (Aσf ) holds, where ∗ (Aσf )(σn)n≥1 ⊂ R+, σn → ∞, and dom f 6= ∅.

(ii) Then there exists α ∈ R+ such that ∗ I := ] − ∞, −α[ ⊂ dom f ⊂ R− ∩ cl I , f is strictly convex and increasing on dom f , and limx→−∞ f (x) = 0 = inf f . Moreover,

0 X 0 X σnx f (x) = f (x) = pnσne ∀x ∈ int(dom f ) = I , n≥1 n n≥1

0 0 f is increasing and continuous on I , limx→−∞ f (x) = 0, and

0 X −σnα lim f (x) = pnσne =: γ ∈ ]0, ∞]. x↑−α n≥1

In particular, ∂f (int(dom f )) = f 0(I ) = ]0, γ[. C. Z˘alinescu Series of convex functions and entropy minimization Proposition 6 (continued) ∗ (iii) Let α, I , γ be as in (ii). Assume that α ∈ R+. Then either (a) dom f = I and γ = ∞, or 0 (b) dom f = cl I and γ = ∞, in which case f−(−α) = γ, P 0 ∂f (−α) = ∅ and the series n≥1 fn(−α) is divergent, or 0 (c) dom f = cl I and γ < ∞, in which case f−(−α) = γ and X f 0(−α) = γ ∈ [γ, ∞[ = ∂f (−α). n≥1 n

C. Z˘alinescu Series of convex functions and entropy minimization Example 7 θ In Proposition 6, set pn = 1 for n ≥ 1. For σn = n (n ≥ 1) with  θ θ > 0 one has dom f = (−∞, 0), for σn = ln n(ln n) (n ≥ 2) with θ ∈ R one has int(dom f ) = (−∞, −1), while for σn = ln(ln n)  θ (n ≥ 2) one has dom f = ∅. Moreover, let σn = ln n(ln n) (n ≥ 2); for θ ∈ (−∞, 1] one has dom f = (−∞, −1), for 0 θ ∈ (1, 2] one has dom f = (−∞, −1] and f−(−1) = ∞, for 0 θ ∈ (2, ∞) one has dom f = (−∞, −1] and f−(−1) < ∞.

Proposition 6 (iii) (c) and Example 7 show that the conclusion of Theorem 3 can be false for x ∈ dom f \ int(dom f ) [even for x ∈ dom(∂f ) \ int(dom f )]. We have examples which show that the condition int(dom f ) 6= ∅ is is essential in Theorem 3.

C. Z˘alinescu Series of convex functions and entropy minimization Conjugate of a countable sum

The natural question is if we could say something about the P conjugate of f = n≥1 fn when f , fn ∈ Λ(E). Theorem 3 is applied to get the last assertion of the following result. Theorem 9 (to be continued) P Let f , fn ∈ Λ(E). Assume that f (x) = n≥1 fn(x) for every x ∈ E. ∗ ∗ ∗ P ∗ ∗ ∗ (i) If (xn )n≥1 ⊂ (dom fn )n≥1 is such that w - n≥1 xn = x ∈ E , P ∗ ∗ then the series n≥1 fn (xn ) has a limit in R and ∗ ∗ P ∗ ∗ f (x ) ≤ n≥1 fn (xn ); in particular,   ∗ ∗ X ∗ ∗ ∗ ∗ ∗ ∗ X ∗ f (x ) ≤ inf fn (xn ) | (xn )n≥1 ⊂ (dom fn )n≥1, x = w - xn n≥1 n≥1

for every x∗ ∈ E ∗, with the usual convention inf ∅ := +∞.

C. Z˘alinescu Series of convex functions and entropy minimization Theorem 9 (continued) ∗ (ii) If x ∈ ∩n≥1 dom fn and (xn)n≥1 ⊂ (∂fn(x))n≥1 is such that ∗ P ∗ ∗ ∗ ∗ w - n≥1 xn = x ∈ E , then x ∈ dom f , x ∈ ∂f (x) and ∗ ∗ P ∗ ∗ f (x ) = n≥1 fn (xn). In particular,   ∗ ∗ X ∗ ∗ ∗ ∗ ∗ ∗ X ∗ f (x ) = min fn (xn ) | (xn )n≥1 ⊂ (dom fn )n≥1, x = w - xn n≥1 n≥1

(iii) Assume that f and fn (n ≥ 1) are continuous on int(dom f ). Then relation above holds for every x∗ ∈ ∂f (int(dom f )). More ∗ ∗ ∗ P ∗ precisely, if x ∈ ∂f (x) for x ∈ int(dom f ) then x = w - n≥1 xn ∗ ∗ ∗ P ∗ ∗ for some (xn )n≥1 ⊂ (∂fn(x))n≥1, and f (x ) = n≥1 fn (xn ).

C. Z˘alinescu Series of convex functions and entropy minimization Of course, the equality in the preceding result is also valid for x∗ ∈ X ∗ \ dom f ∗. So the problem remains to see what is happening for x∗ ∈ dom f ∗ \ ∂f (int(dom f )) .

Taking fk = 0 for k ≥ n + 1 in the preceding result, the conclusion is much weaker than what we know about the conjugate of a finite sum because nothing is said for x∗ ∈ dom f ∗ \ ∂f (int(dom f )) . In the next proposition we give complete descriptions for f ∗, where f is provided in Proposition 6.

C. Z˘alinescu Series of convex functions and entropy minimization Proposition 10 (to be continued)

σnx Let fn(x) := pne for x ∈ R with pn ∈ [1, ∞), 0 < σn → ∞, and P f = n≥1 fn. Assume that condition (Aσf is satisfied, and so I := (−∞, −α) ⊂ dom f ⊂ cl I for some α ∈ R+. (i) Then ∂f (int(dom f )) = (0, γ), where P −σnα ∗ γ := n≥1 pnσne ∈ R+, dom f = R+, and for any u ∈ R+   ∗ X ∗ ∗ X f (u) ≤ inf fn (un) | (un)n≥1 ⊂ (dom fn )n≥1 , u = un < ∞. n≥1 n≥1

∗ (ii) Let u ∈ R+ (= dom f ). Then u ∈ [0, γ] ∩ R if and only if   ∗ X ∗ ∗ X f (u) = min fn (un) | (un)n≥1 ⊂ (dom fn )n≥1 , u = un n≥1 n≥1  X X  = min un(ln(un/pn) − 1) | (un)n≥1 ⊂ R+, u = σnun . n≥1 n≥1

C. Z˘alinescu Series of convex functions and entropy minimization Proposition 10 (continued) More precisely, the minimum above is attained for: un = 0 (n ≥ 1) when u = 0, σnx 0 un = e (n ≥ 1) when u = f (x) with x ∈ I , −σnα ∗ un = e when u = γ (< ∞) (in which case α ∈ R+, 0 −α ∈ dom ∂f = dom f and f−(−α) = γ). ∗ (iii) Let u ∈ R+ (= dom f ). Then   ∗ X ∗ ∗ X f (u) = inf fn (un) | (un)n≥1 ⊂ (dom fn )n≥1 , u = un n≥1 n≥1  X X  = inf un(ln(un/pn) − 1) | (un)n≥1 ⊂ R+, u = σnun . n≥1 n≥1

As for Theorem 3, we have examples which show that the condition int(dom f ) 6= ∅ is essential in Theorem 9 (iii).

C. Z˘alinescu Series of convex functions and entropy minimization Applications to Statistical Physics

Related to the problem mentioned at the beginning of our talk consider the sequence (σn)n≥1 ⊂ R, and set  X X S(u, v) := (un)n≥1 ⊂ + | u = un, v = σnun R n≥1 n≥1

2 for each (u, v) ∈ R . It is clear that S(u, v) = S(tu, tv) for all 2 ∗ (u, v) ∈ R and t ∈ R+, S(u, v) = ∅ if either u < 0 or u = 0 6= v, and S(0, 0) = {(0)n≥1}. We also set Xn ρn := pk k=1 1  2  ηn := min σk | k ∈ 1, n , ηn := max σk | k ∈ 1, n , η1 := inf {σk | n ≥ 1} ∈ [−∞, ∞[, η2 := sup {σk | n ≥ 1} ∈ ]− ∞, ∞];

of course, limn→∞ ρn = ∞ (because pk ≥ 1 for n ≥ 1).

C. Z˘alinescu Series of convex functions and entropy minimization The entropy minimization problem (EMP for short) of Statistical Mechanics and Statistical Physics associated to W ∈ {EBE , EMB , 2 EFD } and (u, v) ∈ R is (EMP) minimize P p W ( un ) s.t. (u ) ∈ S(u, v), u,v n≥1 n pn n n≥1 P Pn where n≥1 βn := limn→∞ k=1 βk when this limit exists in R P and n≥1 βn := ∞ otherwise. With the preceding convention, it is easy to see that αn ≤ βn for n ≥ 1 imply that P P n≥1 αn ≤ n≥1 βn. Remark 13

Note that for (un)n≥1 ∈ S(u, v) one has that Pn uk limn→∞ pk W ( ) exists in: [−∞, 0] when W = EBE , in k=1 pk [−∞, 0] ∪ {∞} when W = EFD , and in [−∞, ∞[ when W = EMB .

C. Z˘alinescu Series of convex functions and entropy minimization The value (marginal) function associated to problems (EMP)u,v is     2 X un HW : R → R, HW (u, v) := inf pnW | (un)n≥1 ∈ S(u, v) , pn n≥1 with the usual convention inf ∅ := ∞. We shall write simply HBE , HMB , HFD when W is EBE , EMB , or EFD , respectively. Therefore,

 2 dom HW ⊂ dom S := (u, v) ∈ R | S(u, v) 6= ∅ ; hence HW (u, v) = ∞ if either u < 0 or u = 0 6= v, and HW (0, 0) = 0. Taking into account that EBE ≤ EMB ≤ EFD , and using Remark 13, we get

HBE ≤ HMB ≤ HFD ,

dom HFD ⊂ dom HMB = dom HBE = dom S.

C. Z˘alinescu Series of convex functions and entropy minimization Proposition 14 (domain and convexity of HW )

The following assertions hold for W ∈ {EMB , EBE , EFD }: (i) The marginal function HW is convex. (ii) Assume that η1 = η2. Then dom HW = dom S = R+ · (1, σ1); ∗ in particular, ri(dom HW ) = R+(1, σ1) 6= ∅ = int(dom HW ). (iii) Assume that η1 < η2 and take n ≥ 2 such that {σk | k ∈ 1, n} is not a singleton. Then for W ∈ {EBE , EMB } one has [ Xn C := + · (1, σk ) ⊂ dom HW = dom S ⊂ cl C, n≥1 k=1 R n [ X ∗ ∗ int(dom HW ) = int C = R+ · (1, σk ) = R+ · ({1} × ]η1, η2[) ; n≥n k=1 n [ X A := [0, pk ] · (1, σk ) ⊂ dom HFD ⊂ cl A, n≥1 k=1 [ Xn int(dom HFD ) = int A = ]0, pk [ · (1, σk ). n≥n k=1

C. Z˘alinescu Series of convex functions and entropy minimization Proposition 15

Consider W ∈ {EBE , EMB , EFD }. (i) Assume that η1 = η2. Then HW (u, v) = −∞ for all ∗ (u, v) ∈ ri(dom HW ) = R+ · (1, σ1). P σnx (ii) Assume that the series n≥1 pne is divergent for every x ∈ R and η1 < η2. Then

HW (u, v) = −∞ ∀(u, v) ∈ int(dom HW ).

The previous result shows the lack of interest of the EMP when the sequence (σn)n≥1 is constant. Also, it gives a hint on the importance of the properties of the function

X σnx f : → , f (x) = pne R R n≥1 established in Propositions 6 and 10.

C. Z˘alinescu Series of convex functions and entropy minimization Let us consider the following functions for W ∈ {EMB , EFD , EBE } :

W 2 W ∗ hn : R → R, hn (x, y) := pnW (x + σny) > 0 (n ≥ 1, x, y ∈ R), 2 X W hW := → , hW := h ; R R n≥1 n we write simply hMB , hFD , hBE instead of hEMB , hEFD , hEBE , W ∗ 2 respectively. Because hn = (pnW ) ◦ An, where An : R → R is ∗ defined by An(x, y) := x + σny [and so Anw = w(1, σn)], ∗  W  ∗ ∗ ∗ hn (u, v) = min {(pnW ) (w) | Anw = (u, v)}  u pnW ( ) if u ≥ 0 and v = σnu, = pn ∞ otherwise,

W ∗ and so hn is strictly convex on its domain. W ∗ The expression of hn in connection with Theorem 9 shows the interest of studying the properties of the functions hW .

C. Z˘alinescu Series of convex functions and entropy minimization Properties of the functions hW

The (convex) conjugates of these functions EBE , EMB , EFD are

∗ t ∗ t EMB (t) = e ∀t ∈ R, EFD (t) = ln(1 + e ) ∀t ∈ R,  t ∗ ∗ − ln(1 − e ) if t ∈ R−, EBE (t) = ∞ if t ∈ R+.

Moreover, for W ∈ {EBE , EMB , EFD } we have that ∂W (u) = {W 0(u)} for u ∈ int(dom W ) and ∂W (u) = ∅ elsewhere; furthermore,

t ∗ 0 e ∗ (W ) (t) = t ∀t ∈ dom W , 1 + aW e where   −1 if W = EBE , aW := 0 if W = EMB ,  1 if W = EFD .

C. Z˘alinescu Series of convex functions and entropy minimization Because pn ≥ 1 for n ≥ 1, we have that

∗ ∗ (x, y) ∈ dom hW ⇒ pnW (x + σny) → 0 ⇒ W (x + σny) → 0

⇔ σny → −∞ ⇔ [y > 0 and σn → −∞] or [y < 0 and σn → ∞].

W 2 Of course, hn (x, y) > 0 for all (x, y) ∈ R , n ≥ 1 and W ∈ {EMB , EFD , EBE }; because for σny → −∞ we have

hEFD (x, y) hEBE (x, y) lim n = lim n = 1, n→∞ EMB n→∞ EMB hn (x, y) hn (x, y) we obtain that

dom hFD = dom hMB ,  2 dom hBE = dom hMB ∩ (x, y) ∈ R | x + σny < 0 ∀n ≥ 1 .

P x+σny x Since hMB (x, y) = n≥1 pne = e f (y), clearly dom hMB = R × dom f . It follows that

dom hFD 6= ∅ ⇔ dom hMB 6= ∅ ⇔ dom hBE 6= ∅ ⇔ dom f 6= ∅.

C. Z˘alinescu Series of convex functions and entropy minimization It is natural to consider the case dom hW 6= ∅; in the sequel, we assume that (Aσf ) holds. Proposition 17 (to be continued)

Assume that (Aσf ) holds, and take α ∈ R+ such that I := ]− ∞, −α[ ⊂ dom f ⊂ cl I . Let W ∈ {EMB , EFD , EBE }. (i) Then hW is convex, lower semicontinuous, positive, and

dom hFD = dom hMB = R × dom f , dom hBE = {(x, y) ∈ R × dom f | x + θ1y < 0} , where θ1 := min{σn | n ≥ 1}.

C. Z˘alinescu Series of convex functions and entropy minimization Proposition 17 (continued)

(ii) hW is differentiable at any (x, y) ∈ int(dom hW ) and

X ex+σny ∇hW (x, y) = pn x+σ y · (1, σn), n≥1 1 + aW e n aW being defined above. Moreover, assume that ∗ (x, −α) ∈ dom hW ; in particular, −α ∈ dom f ⊂ R−. Then

X W ∂hW (x, −α) 6= ∅ ⇐⇒ ∇hn (x, −α) converges n≥1

X x−σnα ⇐⇒ γ := pnσne ∈ R; n≥1

P W 2 if (u, v) := n≥1 ∇hn (x, −α) exists in R , then

∂hW (x, −α) = {u} × [v, ∞[ = {(u, v)} + {0} × R+.

C. Z˘alinescu Series of convex functions and entropy minimization Theorem 18

Let W ∈ {EMB , EFD , EBE } and aW be defined before. Then for W every (x, y) ∈ ∩n≥1 dom hn such that the series P ex+σny pn x+σ y · (1, σn) is convergent [this is the case, for n≥1 1+aW e n 2 example, when (x, y) ∈ int(dom hW )] with sum (u, v) ∈ R , the problem (EMP)u,v has the unique optimal solution  ex+σny  pn x+σny . Moreover, the value of the problem (EMP)u,v 1+aW e n≥1 ∗ ∗ is hW (u, v), that is HW (u, v) = hW (u, v). The result in Theorem 18 is obtained generally using the Lagrange multipliers method (LMM) in a formal way. The complete solution to EMP for the Maxwell–Boltzmann entropy is provided in the next result.

C. Z˘alinescu Series of convex functions and entropy minimization Theorem 19 (to be continued) ∗ Let (pn)n≥1 ⊂ [1, ∞[, (σn)n≥1 ⊂ R+ with σn → ∞, and 2 x+σny hn : R → R be defined by hn(x, y) := pne for n ≥ 1 and P x, y ∈ R; set h = n≥1 hn. Assume that dom h 6= ∅. Clearly, h, hn (n ≥ 1) are convex and

x X σny x 2 h(x, y) = e pne = e f (y) ∀(x, y) ∈ . n≥1 R

Since dom h = R × dom f 6= ∅, using Proposition 16, we have that I := ]− ∞, −α[ ⊂ dom f ⊂ cl I for some α ∈ R+. It follows that 2 int(dom h) = R × I ⊂ ∩n≥1 dom hn = R .

C. Z˘alinescu Series of convex functions and entropy minimization Theorem 19 (to be continued) (i) We have that h is differentiable on int(dom h) and

∂h(int(dom h)) = ∇h(R × I )  2 ∗ = (u, v) ∈ R | u ∈ R+, θ1u < v < θ2u ,

0 where θ1 := min{σn | n ≥ 1}, θ2 := limy↑−α f (y)/f (y) ∈ ]θ1, ∞]; 0 P −σnα θ2 < ∞ ⇔ [−α ∈ dom f , γ := f−(−α) = n≥1 pnσne < ∞.] Moreover, if −α ∈ dom f and γ < ∞ then

x X e (f (−α), γ) = ∇hn(x, −α) n≥1 ∈ ∂h(x, −α) = {ex f (−α)} × [ex γ, ∞[.

C. Z˘alinescu Series of convex functions and entropy minimization Theorem 19 (to be continued) 0 (ii) The function ϕ : I → ]θ1, θ2[, ϕ(y) := f (y)/f (y), is bijective (and increasing), ln f : R → R is convex (even strictly convex and increasing on its domain), and  ∞ if w < θ1,  P ∗  − ln n∈Σ pn if w = θ1, (ln f ) (w) = −1  −1  wϕ (w) − ln f (ϕ (w)) if θ1 < w < θ2,   −αw − ln [f (−α)] if θ2 ≤ w,

∗ where Σ := {n ∈ N | σn = θ1}. ∗  2 Moreover, dom h = (u, v) ∈ R | v ≥ θ1u ≥ 0 and

 u ln u − u + u · (ln f )∗(v/u) if v ≥ θ u > 0, h∗(u, v) = 1 −αv if u = 0 ≤ v.

C. Z˘alinescu Series of convex functions and entropy minimization Theorem 19 (continued) (iii) Take (u, v) ∈ dom h∗. Then   ∗ X un h (u, v) = min un(ln −1) | (un)n≥1 ∈ S(u, v) = H(u, v) n≥1 pn

∗ ∗ iff (u, v) ∈ A := {(0, 0)} ∪ {(u, v) ∈ R+ × R+ | θ1u ≤ v ≤ θ2u}. More precisely, for (u, v) ∈ A the minimum is attained at a unique sequence (un)n≥1 ∈ S(u, v), as follows: (a) (un)n≥1 = (0)n≥1 if P (u, v) = (0, 0); (b) un := pnu/ k∈Σ pk if n ∈ Σ, un := 0 if ∗ ∗ n ∈ N \ Σ provided u ∈ R+ and v = θ1u; x+σny ∗ (c) (un)n≥1 = (pne )n≥1 if u ∈ R+ and θ1u < v < θ2u, where −1 x−σnα y := ϕ (v/u) and x := ln [u/f (y)]; (d) (un)n≥1 = (pne )n≥1 ∗ if θ2 < ∞, u ∈ R+ and v = θ2u, where x := ln [u/f (−α)] . ∗ ∗ Moreover, S(0, v) = ∅ if v ∈ R+, and h (u, v) = H(u, v) whenever 0 < θ2u < v (for θ2 < ∞).

C. Z˘alinescu Series of convex functions and entropy minimization Corollary 20

Consider the sequences (pn)n≥1 ⊂ [1, ∞[, (σn)n≥1 ⊂ R and let H := H(pn) := H(pn) . Then (σn) (σn),EMB

∗ X x+σny x X σny 2 H (x, y) = pne = e pne ∀(x, y) ∈ R . n≥1 n≥1

C. Z˘alinescu Series of convex functions and entropy minimization Note that the case in which the entropy is given by R p T h(u(x)) dµ(x) with u ∈ L (T , µ), and the problem is to minimize the entropy with respect to the constraint Au = b, where p m A : L → R is a continuous linear operator, is treated rigorously by J. M. Borwein and his collaborators in the last 25 years. See Borwein (2012) for a recent survey. In those papers (T , µ) is a finite space. J. M. Borwein says: The infinite horizon case with infinite measure is often more challenging and sometimes the corresponding results are false or unproven. J. M. Borwein sketch the formal approach and points out that “There are two major problems with this free-wheeling formal approach: (1) The assumption that a solution xˆ exists (2) The assumption that the Lagrangian is differentiable” It seems that this is the first rigorous (I hope) presentation of the P maximum entropy when the entropy is given by a sum j∈I h(xj ).

C. Z˘alinescu Series of convex functions and entropy minimization References

Borwein, J. M.: Maximum entropy and feasibility methods for convex and nonconvex inverse problems, Optimization 61 (2012), 1–33. Vall´ee,C.; Z˘alinescu,C.: Series of convex functions: subdifferential, conjugate and applications to entropy minimization, J. Convex Anal. 23(4) (2016). Zheng, X. Y.: A series of convex functions on a , Acta Mathematica Sinica, New Series 14 (1998), 77–84.

C. Z˘alinescu Series of convex functions and entropy minimization Thank you for your attention!

C. Z˘alinescu Series of convex functions and entropy minimization