Convex series of convex functions with applications to Statistical Mechanics

Transcription

1 Convex series of convex functions with applications to Statistical Mechanics Constantin Zălinescu University Alexandru Ioan Cuza Iaşi Faculty of Mathematics Melbourne, MODU2016

2 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 (e j ) j I. The total number of ways (of doing this) is Ω = N! j n j!. The problem is to maximize Ω by keeping fixed E, N, that is n j = N, n j e j = E. j j

3 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 ln Ω N ln N N j (n j ln n j n j ) = N ln N j n j ln n j. Taking the Lagrangian L = N ln N j n j ln n j + α j n j N + β j n j e j E, one finds L n j = 0 for every j, and so n j = e α+βe j with j eα+βe j = N, j e je α+βe j = 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.

4 As examples of (e j ) j are: e j = γ(j ), e j = γj(j + 1) (j N), e ijk = γ(i 2 + j 2 + k 2 ), (i, j, k N). Note that all n j (= e α+βe j ) are (strictly) positive and probably very few (maybe none) are integer. However, from the formulation of the starting problem we must have n j N, and so at most N of them might be non-zero.

5 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. Guénault, 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:

6 Is it possible to prove rigorously that the solution found as above is indeed a solution for the problem min n 1 u n(ln u n 1) s.t. n 1 u n = u, n 1 σ nu n = v? Which are the pairs (u, v) R 2 for which the problem above has optimal solutions? If the problem has not optimal solutions, which is the value of this problem?

7 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, f n Λ(X ). Assume that f (x) = n 1 f n(x) for every x E. If x n 1 dom f n and (x n) n 1 ( f n (x)) n 1 is such that w - n 1 x n = x E, then x dom f, x f (x) and f (x ) = n 1 f n (x n). In particular, { f (x ) = min fn (xn) (xn) n 1 (dom fn ) n 1, x = w - } xn n 1 n 1 Moreover, if f and f n are continuous on int(dom f ), and x f (x) with x int(dom f ), then x = w - n 1 x n for some sequence (x n) n 1 ( f n (x)) n 1.

8 Based on this result we study the problems s.t. i u i = u, minimize i p iw ( u i p i ) i u iε i = v, where p i [1, ) and W is one of the functions E BE, E FD, E MB below: { u ln u (1 + u) ln(1 + u) if u R+, E BE (u) := if u R, { u ln u + (1 u) ln(1 u) if u [0, 1], E FD (u) := if u R \ [0, 1], { u ln u u if u R+, E MB (u) := if u R, with 0 ln 0 := 0 and R + := [0, [, R + := ]0, [, R := R +, R := R +; E BE, E FD, E MB are Bose Einstein, Fermi Dirac and Maxwell Boltzmann (or Shannon) entropies, respectively.

9 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{ x, x f (x) x E} the subdifferential of f is f (x) := {x E x x, x f (x ) 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 f +(x, u) := lim t 0+ t 1 [f (x + tu) f (x)] for u E; if f Λ(E) then f +(x, u) exists in R for all x dom f and u E For (A i ) i I a family of (nonempty sets), (a i ) i I (A i ) i I means that a i A i for all i I

10 Definition (Zheng, 1998) Let A, A n P 0 (E) := {F E F } (n 1). One says that (A n ) n 1 converges normally to A (wrt τ), written A = τ- n 1 A n, if: (I) for every sequence (x n ) n 1 (A n ) n 1, the series n 1 x n τ-converges and its sum x belongs to A; (II) for each (τ-)neighborhood U of 0 in E (that is U N τ E ) there is n 0 1 such that τ- k n x k U for all sequences (x n ) n 1 (A n ) n 1 and all n n 0 (observe that the series k n x k is τ-convergent by (I)); (III) for each x A there exists (x n ) n 1 (A n ) n 1 such that x = τ- n 1 x n. Observe that A in the above definition is unique; moreover, A is convex if all A n are convex.

11 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 ) Let f, f n Λ(E) be such that f (x) = n 1 f n(x) for every x E. Assume that x core(dom f ). Then f +(x, u) = n 1 f n+(x, u) u E.

12 Theorem 2 (uniform convergence) Let f, f n Λ(E) be such that f (x) = n 1 f n(x) for every x E. Assume that the series n 1 f n converges uniformly on a neighborhood of x 0 int(dom f ). Then for every x int(dom f ) there exists a neighborhood U NE τ with x + U dom f such that the series n 1 f n+(, ) converges uniformly [to f +(, )] on (x + U) U.

13 Theorem 3 (formula for the subdifferential of f ) Let f, f n Λ(E). Assume that f (x) = n 1 f n(x) for every x E. If f and f n are continuous on int(dom f ), then f (x) = w - n 1 f n (x) x int(dom f ).

14 When E is a normed vector space, one has also the next result. Theorem 4 Let E be a normed vector space and f, f n Λ(E). Assume that f (x) = n 1 f n(x) for every x E. If f and f n are continuous on int(dom f ) and the series n 1 f n converges uniformly on a nonempty open subset of dom f, then f (x) = - n 1 f n (x) x int(dom f ); moreover, lim n k n f k(x) = 0 uniformly on some neighborhood of x for every x int(dom f ), where A := sup { x x A} for A X.

15 Corollary 5 Let f, f n Λ(E). Assume that f (x) = n 1 f n(x) for every x E, and f, f n are continuous on int(dom f ) for every n 1. Take x int(dom f ). Then (i) f is Gâteaux differentiable at x if and only if f n is Gâteaux differentiable at x for every n 1. (ii) Moreover, assume that E is a normed vector space. If f is Fréchet differentiable at x then f n is Fréchet differentiable at x for every n 1.

16 In particular, f (int(dom f )) = f (I ) = ]0, γ[. Proposition 6 (to be continued) Let f n (x) := p n e σnx with p n 1 for n 1, x R; f = n 1 f n. (i) If x dom f then σ n x, and so either x > 0 and σ n, or x < 0 and σ n. Furthermore, assume that (A σf ) holds, where (A σf ) (σ n ) n 1 R +, σ n, and dom f. (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 lim x f (x) = 0 = inf f. Moreover, f (x) = n 1 f n(x) = n 1 p nσ n e σnx x int(dom f ) = I, f is increasing and continuous on I, lim x f (x) = 0, and lim f (x) = p nσ n e σnα =: γ ]0, ]. x α n 1

17 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 (b) dom f = cl I and γ =, in which case f ( α) = γ, f ( α) = and the series n 1 f n( α) is divergent, or (c) dom f = cl I and γ <, in which case f ( α) = γ and n 1 f n( α) = γ [γ, [ = f ( α).

18 Example 7 In Proposition 6, set p n = 1 for n 1. For σ n = n θ (n 1) with θ > 0 one has dom f = (, 0), for σ n = ln [ n(ln n) θ] (n 2) with θ R one has int(dom f ) = (, 1), while for σ n = ln(ln n) (n 2) one has dom f =. Moreover, let σ n = ln [ n(ln n) θ] (n 2); for θ (, 1] one has dom f = (, 1), for θ (1, 2] one has dom f = (, 1] and f ( 1) =, for 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 ) is is essential in Theorem 3.

19 Conjugate of a countable sum The natural question is if we could say something about the conjugate of f = n 1 f n when f, f n Λ(E). Theorem 3 is applied to get the last assertion of the following result. Theorem 9 (to be continued) Let f, f n Λ(E). Assume that f (x) = n 1 f n(x) for every x E. (i) If (xn) n 1 (dom fn ) n 1 is such that w - n 1 x n = x E, then the series n 1 f n (xn) has a limit in R and f (x ) n 1 f n (xn); in particular, { 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 := +.

20 Theorem 9 (continued) (ii) If x n 1 dom f n and (x n) n 1 ( f n (x)) n 1 is such that w - n 1 x n = x E, then x dom f, x f (x) and f (x ) = n 1 f n (x n). In particular, { f (x ) = min fn (xn) (xn) n 1 (dom fn ) n 1, x = w - } xn n 1 n 1 (iii) Assume that f and f n (n 1) are continuous on int(dom f ). Then relation above holds for every x f (int(dom f )). More precisely, if x f (x) for x int(dom f ) then x = w - n 1 x n for some (x n) n 1 ( f n (x)) n 1, and f (x ) = n 1 f n (x n).

21 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 f k = 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.

22 Proposition 10 (to be continued) Let f n (x) := p n e σnx for x R with p n [1, ), 0 < σ n, and f = n 1 f n. Assume that condition (A σf is satisfied, and so I := (, α) dom f cl I for some α R +. (i) Then f (int(dom f )) = (0, γ), where γ := n 1 p nσ n e σnα R +, dom f = R +, and for any u R + { f (u) inf fn (u n ) (u n ) n 1 (dom fn ) n 1, u = } u n <. n 1 n 1 (ii) Let u R + (= dom f ). Then u [0, γ] R if and only if { f (u) = min fn (u n ) (u n ) n 1 (dom fn ) n 1, u = } u n n 1 n 1 { = min σ n u n }. n 1 u n (ln(u n /p n ) 1) (u n ) n 1 R +, u = n 1

23 Proposition 10 (continued) More precisely, the minimum above is attained for: u n = 0 (n 1) when u = 0, u n = e σnx (n 1) when u = f (x) with x I, u n = e σnα when u = γ (< ) (in which case α R +, α dom f = dom f and f ( α) = γ). (iii) Let u R + (= dom f ). Then { f (u) = inf fn (u n ) (u n ) n 1 (dom fn ) n 1, u = } u n n 1 n 1 { = inf σ n u n }. n 1 u n (ln(u n /p n ) 1) (u n ) n 1 R +, u = n 1 As for Theorem 3, we have examples which show that the condition int(dom f ) is essential in Theorem 9 (iii).

24 Applications to Statistical Physics Related to the problem mentioned at the beginning of our talk consider the sequence (σ n ) n 1 R, and set S(u, v) := { (u n ) n 1 R + u = n 1 u n, v = n 1 σ nu n } for each (u, v) R 2. It is clear that S(u, v) = S(tu, tv) for all (u, v) R 2 and t R +, S(u, v) = if either u < 0 or u = 0 v, and S(0, 0) = {(0) n 1 }. We also set ρ n := n p k k=1 ηn 1 := min { σ k k 1, n }, ηn 2 := max { σ k k 1, n }, η 1 := inf {σ k n 1} [, [, η 2 := sup {σ k n 1} ], ]; of course, lim n ρ n = (because p k 1 for n 1).

25 The entropy minimization problem (EMP for short) of Statistical Mechanics and Statistical Physics associated to W {E BE, E MB, E FD } and (u, v) R 2 is (EMP) u,v minimize n 1 p nw ( un p n ) s.t. (u n ) n 1 S(u, v), where n 1 β n := lim n n k=1 β k when this limit exists in R and n 1 β n := otherwise. With the preceding convention, it is easy to see that α n β n for n 1 imply that n 1 α n n 1 β n. Remark 13 Note that for (u n ) n 1 S(u, v) one has that lim n n k=1 p kw ( u k p k ) exists in: [, 0] when W = E BE, in [, 0] { } when W = E FD, and in [, [ when W = E MB.

26 The value (marginal) function associated to problems (EMP) u,v is { ( ) } H W : R 2 un R, H W (u, v) := inf p n W (u n ) n 1 S(u, v), n 1 with the usual convention inf :=. We shall write simply H BE, H MB, H FD when W is E BE, E MB, or E FD, respectively. Therefore, dom H W dom S := { (u, v) R 2 S(u, v) } ; hence H W (u, v) = if either u < 0 or u = 0 v, and H W (0, 0) = 0. Taking into account that E BE E MB E FD, and using Remark 13, we get H BE H MB H FD, dom H FD dom H MB = dom H BE = dom S. p n

27 Proposition 14 (domain and convexity of H W ) The following assertions hold for W {E MB, E BE, E FD }: (i) The marginal function H W is convex. (ii) Assume that η 1 = η 2. Then dom H W = dom S = R + (1, σ 1 ); in particular, ri(dom H W ) = R +(1, σ 1 ) = int(dom H W ). (iii) Assume that η 1 < η 2 and take n 2 such that {σ k k 1, n} is not a singleton. Then for W {E BE, E MB } one has n C := R + (1, σ k ) dom H W = dom S cl C, n 1 k=1 int(dom H W ) = int C = n R + (1, σ k ) = R + ({1} ]η 1, η 2 [) ; n n A := n 1 k=1 n [0, p k ] (1, σ k ) dom H FD cl A, k=1 int(dom H FD ) = int A = n n n k=1 ]0, p k[ (1, σ k ).

28 Proposition 15 Consider W {E BE, E MB, E FD }. (i) Assume that η 1 = η 2. Then H W (u, v) = for all (u, v) ri(dom H W ) = R + (1, σ 1 ). (ii) Assume that the series n 1 p ne σnx is divergent for every x R and η 1 < η 2. Then H W (u, v) = (u, v) int(dom H W ). The previous result shows the lack of interest of the EMP when the sequence (σ n ) n 1 is constant. Also, it gives a hint on the importance of the properties of the function f : R R, f (x) = n 1 p ne σnx established in Propositions 6 and 10.

29 Let us consider the following functions for W {E MB, E FD, E BE } : h W n : R 2 R, h W n (x, y) := p n W (x + σ n y) > 0 (n 1, x, y R), h W := R 2 R, h W := n 1 hw n ; we write simply h MB, h FD, h BE instead of h EMB, h EFD, h EBE, respectively. Because hn W = (p n W ) A n, where A n : R 2 R is defined by A n (x, y) := x + σ n y [and so A nw = w(1, σ n )], ( ) hn W (u, v) = min {(pn W ) (w) A nw = (u, v)} { pn W ( u = p n ) if u 0 and v = σ n u, otherwise, and so ( hn W ) is strictly convex on its domain. The expression of ( hn W ) in connection with Theorem 9 shows the interest of studying the properties of the functions h W.

30 Properties of the functions h W The (convex) conjugates of these functions E BE, E MB, E FD are E MB (t) = et t R, E FD (t) = ln(1 + et ) t R, E BE (t) = { ln(1 e t ) if t R, if t R +. Moreover, for W {E BE, E MB, E FD } we have that W (u) = {W (u)} for u int(dom W ) and W (u) = elsewhere; furthermore, (W ) (t) = e t 1 + a W e t t dom W, where 1 if W = E BE, a W := 0 if W = E MB, 1 if W = E FD.

31 Because p n 1 for n 1, we have that (x, y) dom h W p n W (x + σ n y) 0 W (x + σ n y) 0 σ n y [y > 0 and σ n ] or [y < 0 and σ n ]. Of course, h W n (x, y) > 0 for all (x, y) R 2, n 1 and W {E MB, E FD, E BE }; because for σ n y we have we obtain that h E FD n (x, y) lim n h E MB n (x, y) = lim n h E BE n (x, y) dom h FD = dom h MB, h E MB n (x, y) = 1, dom h BE = dom h MB { (x, y) R 2 x + σ n y < 0 n 1 }. Since h MB (x, y) = n 1 p ne x+σny = e x f (y), clearly dom h MB = R dom f. It follows that dom h FD dom h MB dom h BE dom f.

32 It is natural to consider the case dom h W ; in the sequel, we assume that (A σf ) holds. Proposition 17 (to be continued) Assume that (A σf ) holds, and take α R + such that I := ], α[ dom f cl I. Let W {E MB, E FD, E BE }. (i) Then h W is convex, lower semicontinuous, positive, and dom h FD = dom h MB = R dom f, dom h BE = {(x, y) R dom f x + θ 1 y < 0}, where θ 1 := min{σ n n 1}.

33 Proposition 17 (continued) (ii) h W is differentiable at any (x, y) int(dom h W ) and e x+σny h W (x, y) = p n n a W e x+σny (1, σ n), a W being defined above. Moreover, assume that (x, α) dom h W ; in particular, α dom f R. Then h W (x, α) n 1 h W n (x, α) converges γ := n 1 p n σ n e x σnα R; if (u, v) := n 1 hw n (x, α) exists in R 2, then h W (x, α) = {u} [v, [ = {(u, v)} + {0} R +.

34 Theorem 18 Let W {E MB, E FD, E BE } and a W be defined before. Then for every (x, y) n 1 dom hn W such that the series n 1 p e x+σny n 1+a W e (1, σ x+σny n ) is convergent [this is the case, for example, when (x, y) int(dom h W )] with sum (u, v) R 2, the problem ( (EMP) u,v has the unique optimal solution p n e x+σny 1+a W e )n 1. Moreover, the value of the problem (EMP) x+σny u,v is hw (u, v), that is H W (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.

35 Theorem 19 (to be continued) Let (p n ) n 1 [1, [, (σ n ) n 1 R + with σ n, and h n : R 2 R be defined by h n (x, y) := p n e x+σny for n 1 and x, y R; set h = n 1 h n. Assume that dom h. Clearly, h, h n (n 1) are convex and h(x, y) = e x n 1 p ne σny = e x f (y) (x, y) R 2. Since dom h = R dom f, using Proposition 16, we have that I := ], α[ dom f cl I for some α R +. It follows that int(dom h) = R I n 1 dom h n = R 2.

36 Theorem 19 (to be continued) (i) We have that h is differentiable on int(dom h) and h(int(dom h)) = h(r I ) = { (u, v) R 2 u R +, θ 1 u < v < θ 2 u }, where θ 1 := min{σ n n 1}, θ 2 := lim y α f (y)/f (y) ]θ 1, ]; θ 2 < [ α dom f, γ := f ( α) = n 1 p nσ n e σnα <.] Moreover, if α dom f and γ < then e x (f ( α), γ) = h n(x, α) n 1 h(x, α) = {e x f ( α)} [e x γ, [.

37 Theorem 19 (to be continued) (ii) The function ϕ : I ]θ 1, θ 2 [, 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 < θ 1, (ln f ) ln (w) = n Σ p n if w = θ 1, wϕ 1 (w) ln [ f (ϕ 1 (w)) ] if θ 1 < w < θ 2, αw ln [f ( α)] if θ 2 w, where Σ := {n N σ n = θ 1 }. Moreover, dom h = { (u, v) R 2 v θ 1 u 0 } and h (u, v) = { u ln u u + u (ln f ) (v/u) if v θ 1 u > 0, αv if u = 0 v.

38 Theorem 19 (continued) (iii) Take (u, v) dom h. Then { h (u, v) = min u n(ln u } n 1) (u n ) n 1 S(u, v) = H(u, v) n 1 p n iff (u, v) A := {(0, 0)} {(u, v) R + R + θ 1 u v θ 2 u}. More precisely, for (u, v) A the minimum is attained at a unique sequence (u n ) n 1 S(u, v), as follows: (a) (u n ) n 1 = (0) n 1 if (u, v) = (0, 0); (b) u n := p n u/ k Σ p k if n Σ, u n := 0 if n N \ Σ provided u R + and v = θ 1 u; (c) (u n ) n 1 = (p n e x+σny ) n 1 if u R + and θ 1 u < v < θ 2 u, where y := ϕ 1 (v/u) and x := ln [u/f (y)]; (d) (u n ) n 1 = (p n e x σnα ) n 1 if θ 2 <, u R + and v = θ 2 u, where x := ln [u/f ( α)]. Moreover, S(0, v) = if v R +, and h (u, v) = H(u, v) whenever 0 < θ 2 u < v (for θ 2 < ).

39 Corollary 20 Consider the sequences (p n ) n 1 [1, [, (σ n ) n 1 R and let H := H (pn) (σ n) := H(pn) (σ n),e MB. Then H (x, y) = n 1 p n e x+σny = e x n 1 p n e σny (x, y) R 2.

40 Note that the case in which the entropy is given by T h(u(x)) dµ(x) with u Lp (T, µ), and the problem is to minimize the entropy with respect to the constraint Au = b, where A : L p R m 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 measure 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 ˆx exists (2) The assumption that the Lagrangian is differentiable It seems that this is the first rigorous (I hope) presentation of the maximum entropy when the entropy is given by a sum j I h(x j).

41 References Borwein, J. M.: Maximum entropy and feasibility methods for convex and nonconvex inverse problems, Optimization 61 (2012), Vallée, C.; Zălinescu, 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 Banach space, Acta Mathematica Sinica, New Series 14 (1998),

42 Thank you for your attention!