Convex series 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 derivative 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,