Lagrangian submanifolds generated by the Maximum Entropy principle

Abstract

We show that the Maximum Entropy principle (E.T. Jaynes, [8]) has a natural description in terms of Morse Families of a Lagrangian submanifold. This geometric approach becomes useful when dealing with the M.E.P. with nonlinear constraints. Examples are presented using the Ising and Potts models of a ferromagnetic material.

1 A synopsis of Morse families

Let M be a smooth even–dimensional manifold and ω a non–degenerate closed two–form. The couple (M, ω) will be called a symplectic manifold. A submanifold S ⊂ M such that 2dimS = dimM and that ω_|S = 0 is a Lagrangian submanifold of M.

As an example, given a manifold Q, its cotangent bundle T*Q equipped with the two–form ω = d_p ∧ d_q, where (q, p) is a local chart of T*Q, is a symplectic manifold. A function G : $Q\to \mathbb{R}$ defines a Lagrangian submanifold of T*Q

(dimΛ_G = dimQ and d(dG ∧ d_q) = 0) that is transversal to the fibers of π : T*Q → Q since π_|ΛG is a diffeomorphism.

Let us consider a family of functions over Q depending on k parameters

and let us denote with

the critical set (the set of critical points over the fibers of Q × U → Q).

The above introduced family is a Morse family (see e.g. Weinstein, [12]) of functions if ε is a regular set in Q × U , i.e. iff the following local condition hold on ε

(1)

where G_u denotes partial derivative of G with respect to u.

Proposition 1.1

(Weinstein, [12] Lect. 6) Let G be a Morse family. Then

is a Lagrangian submanifold1 of T * Q, generated by G.

If the above rank condition defining a Morse family is satisfied by the square matrix G_uu, i.e.

(2)

then, by the implicit function theorem, locally at q, ε is the graph of a function g_W: W ⊂ Q → U, u = g_W (q) and setting one has that

(3)

Proposition 1.2

If the above condition (2) holds we say that all the parameters u can be eliminated. Hence, on W, Λ ∩ T*W coincides with graph therefore Λ is a Lagrangian submanifold locally transversal to the fibers of π.

If G is a Morse family, then the set Γ ⊂ Q of the points q ∈ Q such that at (q, p) ∈ Λ the lagrangian submanifold Λ is not transversal to the fibers of π is called the caustic of Λ and it is given by

(4)

1.1 An example: Lagrange multipliers method

We restate the classical Lagrange multipliers method for the constrained extremization of a differentiable function in terms of Morse families.

Let M be a n–dimensional smooth manifold and f be a smooth real function defined on it. Let , k ≤ n be a smooth submersion i.e. verifying

As it is well known, m* ∈ M is an extremal point, df (m*) = 0, of f on the fiber ζ⁻¹(c), c ∈ if and only if, upon introducing the smooth map

the couple (m*,λ) satisfies the system of equations (gradient system)

(5)

Note that, if the hypothesis rkdζ = max holds, by (5)₂ we can associate to the constrained extremum m* a uniquely defined Lagrange multiplier λ*.

It is straightforward to realize that the set of solutions of the above system of equations (5) when c varies in Range(ζ) has the structure of a critical set with respect to the projection (c, λ, m) ↦ c, i.e.

(6)

Moreover, it is easy to see that the condition that G be a Morse family reads u = (λ, m)

where I_kis the identity matrix in and G_mm ∈ Sym(n) is the Hessian matrix of G with respect to the m variables (summation over repeated indices is understood)

(7)

By Proposition 1.1, if G is a Morse family, the set

(8)

is a Lagrangian submanifold of .

The local condition (2) for the elimination of all the parameters is, in this setting,

(9)

A sufficient condition (not necessary in the general case) for the elimination of all the parameters is given by the following Proposition (see Berteskas,[3] p. 69)

Proposition 1.3

If the symmetric matrix G_mm(λ, m) ∈ Sym(n) in (7) is (positive or negative) definite on ker for (c,λ, m) ∈ ε, then the square matrix G_uu in (9) has maximal rank, and hence all the parameters u = (λ, m) can be eliminatedand the Lagrangian submanifold in (8) is, locally at c, transversal to the fibers of π.

Proof.

We have to show that G_uuz = 0 implies that z = 0, where . Now, setting , we have that

Now, if v = 0, the equation implies that w = 0 since dζ^T is injective (dζ is surjective by hypothesis) and we are done. As a reductio ad absurdum hypothesis, suppose that v ≠ 0 and dζv = 0, i.e. v ∈ ker dζ. The equation dζv = 0 implies v^Tdζ^T = 0^T; by multiplying the second equation by v^T/ ≠ 0 we get

which is an absurdum since G_mm is a (positive on negative) definite matrix on ker dζ(m). ☐

If all the parameters can be eliminated, then, locally at c, the critical set ε is the graph of a function and, since (5)₁ holds, locally at c, the Morse family reduces to

(10)

Remark.

The above sufficient condition involves the ”second order” (Hessian) matrix Gmm in (7). In case ζ is a linear fibration, the above condition is satisfied if Hess(f) is a definite matrix on ker dζ(m).

Example.

In case that f is the potential energy of a mechanical system subject to conservative forces, M is the space of configurations of the system, and ζ: is the fibration defining an ideal (workless) constraint, the above introduced Lagrangian submanifold Λ is the set of couples (c, −λ) respectively value of the constraint and reaction force of the constraint in the equilibrium configuration m. If for every given c there is a unique equilibrium configuration then the value of the reaction force is a function of c. This situation is an example of the case where all the parameters can be globally eliminated.

1.2 Global transversality forΛ

The above condition (2) states that the set of critical points (m, λ) over the fiber ζ⁻¹(c) is made of isolated points. If the condition of the Proposition 1.3 above hold, then all the critical points are local maxima (or minima) of f. We now look at the hypothesis that imply the existence of a unique maximum for f on ζ⁻¹, so that the Lagrangian submanifold Λ is globally transversal to the fibers of π.

A sufficient condition is the following (see [2], p.136):

Proposition 1.4

If the matrix G_uu is positive definite on some bounded convex domain D, then the map G_u is globally univalent on D, hencethe equation G_u = 0 has at most one solution in D.

2 The M. E. P. with linear constraints.

The Maximum Entropy principle (MEP) (Jaynes, [8]) is a general method of statistical inference that it is able to single out a unique probability distribution over the space of possible states i = 1,…,n of a system (to be considered as outcomes of n independent, identically distributed variables) among those compatibles with the results of measurements made on the system, usually in the form of average values of observables defined for the system.

Let χ = {1, 2,…, n} be the (finite) set of possible states (outcomes) of a system. As an example, one can think i = (q_i, p_i) and χ is a finite discretization in cells of the phase space of an Hamiltonian system.

Let be the value assumed by the observable2 ϕ_L : on the state i ∈ χ, L = 1,…, k ≤ n.

The n × k matrix defines a linear fibration from if the observables are independent, i.e.

(11)

A statistical state over χ (an ensemble in the Statistical Mechanics language) is a probability vector p = (p₁,…, p_n) ∈ ; the (information) entropy of p is given by the following function (entropy) defined on

(12)

Note that S is a convex function and that the Hessian of S is

We will call the space of microscopic states of the system and the space of macroscopic states.

Remark.

 In the sequel, the fact that the linear operator ϕ will be restricted to the manifold with boundary will play no role since all the critical points of S will be in the interior of , where .

(M.E.P.) The selected probability distribution3 p is the (unique) one maximizing the information entropy (uncertainty)

among those verifying

(13)

Now we show that the M.E.P. procedure can be recast in the scheme developed in the last Section. (See also [9], for a different approach of M.E.P. in the framework of Symplectic Geometry).

The above constrained maximization problem (M.E.P.) can be dealt with by the Lagrange Multipliers Method by introducing the function

Note that the apparatus of the Lagrange multipliers method supplemented by second order sufficient conditions of Proposition 1.3 and Proposition 1.4 determines the constrained local maximizers of the entropy function. The main feature of the linear constraint case is that there is a unique maximizer.

As a straightforward consequence of Proposition 1.3 (with m = p and G_mm = −H_S) we have the following result:

Proposition 2.1

Since ϕ defines a (linear) fibration and −HS(p) ∈ Sym⁺(n) is a positive definite matrix, then

i) 

condition (1) holds, hence S(c; λ, p) is a Morse family where

ii) 

all the k + n parameters (λ, p) canbe eliminated; hence

iii) 

Λ in (8) is a Lagrangian submanifold locally transversal to the fibers of π.

As we have seen in Proposition 1.2 the critical set ε

(14)

is, locally at c, the graph of a map

such that

Now, setting – see (10)– we have that

(15)

and, by (3) and Proposition 1.2, that

since (c, g(c)) ∈ ε. Hence the projected submanifold Λ is, locally at W , the graph of

(16)

It is instructive to look at the explicit form of the local map g that allows for the elimination of the extra parameters u = (λ, p). The map g has to be determined by the equations in (14). More in detail (here summation over repeated indices is understood):

(17)

Note that we do not take into account explicitly the normalization constraint. From the second equation we get

(18)

which is injective since ϕ^T is injective by hypothesis (11); hence we have succeeded to give the parameters p as a function of the λ. Moreover, by substituting (18) in the first line of (17) we get the following system of k equations in the k unknowns λ_L:

(19)

where is defined as follows:

If the gradient map in (19) has an (at least local) inverse , then by substitution in (18) we get and we are done. The existence of a local inverse is granted –through the implicit function theorem– by Proposition 2.1 which is the reformulation of Proposition 1.3 to the M.E.P. setting. However, the existence can be checked directly given the explicit form of g as follows: a sufficient condition for local invertiblity of (19) is that the matrix

be non-singular.

The global invertiblity can be investigated using Proposition 1.4: a sufficient condition for global invertiblity of the map (19) on a bounded convex domain is that the matrix be (positive or negative) definite. But, for ,

from which we get

hence and we have proved the global invertiblity therefore Λ is globally transversal to the fibers of π.

Of course, an explicit formula for the inverse of the gradient map (19) does not exists.

Let us denote with the inverse of (19). Then the generating function in (15) of the projected lagrangian submanifold (16) has the form

Remark 1.

The above relation between is the same as the one between the Lagrangian and the Hamiltonian H(p).

Remark 2.

Legendre transform. It is now easy to show that the inverse of a gradient map, if it exists, it is a gradient map too. In fact we have

(20)

Remark 3.

It is interesting to consider the case that the matrix ϕ depends smoothly on α real parameters (a₁,…, a_α) ∈ A ⊂ ℝ^α, that is

As a consequence , the above Legendre transform formula inherits a dependence on the parameters a

3 Examples of M.E.P. with nonlinear constraints

In the case of M.E.P. with linear constraints we have seen that the Lagrangian submanifold associated to the constrained maximization problem is always (globally) transversal to the fibers of π, hence the M.E.P. singles out a unique statistical state. We will see that this feature is no longer conserved in case of M.E.P. with nonlinear constraints.

We point out some of the relevant features of the nonlinear case through some examples and remarks.

Example 1.

Let χ = {1,…, n} be the space state of a system and suppose that the observable ϕ satisfy the hypothesis : there exists i, j ∈ χ such that ϕ_i > 0, and ϕ_j < 0. Then for every c > 0 there are solutions p_i > 0 to the constraint equation

or

Now we suppose that the result of an experiment gives the value of 〈ϕ〉²; therefore we are lead to find the maximum entropy probability distribution subject to the nonlinear constraint

The solutions to this constrained maximization problem are the union of the solution (which are unique) to the maximum entropy problem with linear constraints (+) and (−). These latter can be found as explained in Sect.2 and are

Hence, for every c > 0 there are exactly two maximum entropy distributions, and the critical set is

The associated Lagrangian submanifold is given by equation

It is easy to see that Λ has the structure of the graph of a two–valued function defined for c > 0. Note that for c = 0 the two constraints (+) and (−) do coincide, therefore the above function is single–valued. However, for c = 0 the constraint ζ(p) = 〈ϕ〉² fails to have maximal rank on the set of solutions of ζ(p) = c = 0. The following ones are physical examples that can be easily recast into this scheme.

Example 2.

A physical instance of the M.E.P. with nonlinear constraints is the Curie–Weiss approximation of the Ising model for ferromagnetism that we briefly describe below (see e.g.[4]). Let be a lattice containing N sites each occupied by an atom whose spin can assume the value s = +1 or s = −1. The set of possible spin configurations of the lattice is . The potential energy of the lattice (ferromagnet) in the configuration ω ∈ Ω and subject to an external magnetic field B is defined as

(21)

where [i, j] is a couple of nearest–neighborhood sites. We are therefore in the case of an observable (the energy) depending on a parameter (the field strength B). The Curie–Weiss approximation of this model is to neglect the interactions between neighboring spins and hence to assume that the values of the spin at the different sites i = 1,… ,n are independent identically distributed random variables taking values in {−1, +1}; therefore

The average value of the energy of the lattice is

where

Here p+, p− is the probability of the spin up, spin down configuration respectively for the spin at an arbitrary site. Therefore the linear constraint on the lattice energy

where H_min ≤ E ≤ H_max, is equivalent to the nonlinear constraint on 〈s〉 = p₊ − p₋

(22)

for the average value of the spin on a single site.

From now on we consider the space state to be {1,−1} and we look for the probability distribution maximizing the entropy (12) subject to the normalization and energy constraints. In order to apply the theory developed in the previous Sections we are lead to introduce the normalization constraint in the general form ) in this setting the average value of the spin is

The constraint equation are therefore, see (22)

(23)

(24)

with

In this rather special example (with k = n, the number of constraints equal to the dimension of the microscopic states manifold) the fiber , c = (α, γ) is made of isolated points; these are the intersections in the positive quarter plane of the line with the parabola There can be none, one or two intersection points.

Example 3.

Planar Potts model. A straightforward generalization of the Ising model is the planar Potts model. Here the spin vector can point in q ≥ 2 directions in the plane, each forming an angle with a fixed reference axis in the plane. The set of possible spin configurations of the lattice is . The potential energy of the lattice (ferromagnet) in the configuration ω ∈ Ω and subject to an external magnetic field B is

where [i, j] is a couple of nearest–neighborhood sites, dot denotes scalar product and n is the versor of the fixed reference axis. In the Curie–Weiss approximation of statistical independence of the state s_i with respect to s_j (i.e. there is no correlation between the spin at neighboring sites) the average value of the lattice energy is given by

(25)

where the superscript ^T denotes transposition, A ∈ Sym(q) is the interaction symmetric matrix

and has components

The spectrum of the matrix A has the following expression

therefore equation (25) defines an hypercylinder in ℝq. The fiber ζ⁻¹(c), is the intersection, in the positive octant, of the normalization constraint hyperplane

with the (convex) energy hypercylinder (25).

4 Phase transitions in the M.E.P. setting

We consider, as in the previous examples, a physical system in which, for simplicity’ sake, the only observable of interest is the energy, and we suppose that the information we have on the system energy E places a (possibly nonlinear) constraint on the probability distribution

Then the set of macroscopic states of the system, determined by the M.E.P., is the Lagrangian submanifold –see (8)–

where β is the Lagrangian multiplier associated to the energy constraint that can be given in the linear case –see (20) and (15)–as

(26)

In case Λ is transversal to the base manifold, then there is a unique β associated to E; if Λ is in general position w.r.t. the base manifold, then there can be multiple values β associated to the system energy E. This multi–valuedness is interpreted as a phase transition for the system. Note that, as developed in the previous Sections, phase transitions are possible within this scheme only if nonlinear constraints are in force.

Critical values of E, where transitions can occur, can be (in principle) determined through formula (4). However, (4) requires the computation of the solution of a nonlinear system of (n + k) ≈ n equations and of the determinant of a order ≈ n matrix. Therefore (4) gives both an experimentally testable quantity (the critical values of E or the related temperature β) and sets the limits of applicability of this formalism in terms of system complexity.

Finally, let us mention briefly some other approaches to the existence of phase transitions for discrete systems with finite volume (i.e. not in the thermodynamic limit).

For some Hamiltonian systems the entropy of the system having energy E can be, explicitly or at least numerically, computed through the Boltzmann formula

where W (E) is the area in phase space of the E–level set of the Hamiltonian H(q, p); the system temperature is defined as (compare with the above formula (26))

If S(E) is not a convex function of E on its domain, i.e.

where C_v is the specific heat at constant volume, then there may be multiple values of the energy for a given value of the temperature, which is again interpreted with the presence of a phase transition for the system.

This approach (which dates back to Boltzmann) has ben applied in literature to self–gravitating system ([10], [11]), to the Potts models ([7]), and to mechanical systems with non convex potential energy ([1], [5]). For systems with an Hamiltonian of the form kinetic plus potential energy, the phenomenon of negative specific heat stems from the fact that in some energy intervals, an increase of the total energy E determines an increase of the average system’ potential energy and a diminution of the average kinetic energy (i.e. the temperature of the system).