On applications of Orlicz Spaces to Statistical Physics

We present a new rigorous approach based on Orlicz spaces for the description of the statistics of large regular statistical systems, both classical and quantum. This approach has the advantage that statistical mechanics is much better settled. In particular, a new kind of renormalization leading to states having a well defined entropy function is presented.

• canonical quantization; no via bounded operators -so Hilbert space should be of infinite dimension. Consequently, "quantum" is not equivalent to "non-commutative" -a description of a quantum system should be compatible with the canonical quantization. In particular, M n (C), is well suited for a description of a non-commutative system, BUT not for a quantum one.
• individual characteristic of a system will be taken into account -Haag-Kastler approach.
• Consequently, non-commutative calculus, stemming from C * -algebra theory, will be used and all basic features of quantization will be taken into account.

On applications of Orlicz spaces
Applied Math & Math Methods in Physics, Gdansk, December 12, 2013 • Classical statistical physics • Maxwell, Boltzmann -laid the base for statistical mechanics (classical).
• to select a function f which can describe a probability (velocity distribution function) it was necessary to assume: f dx = 1 < ∞; consequently L 1 space was appeared and subsequently, to have a dual pair (observables and states) the pair of Banach spaces < L ∞ , L 1 > was also appeared.
• Boltzmann theory: model of gas, Boltzmann equation, H-theorem; there are no paradoxes -cf Kac explanations.
• Let me term by "Boltzmann dream" the following problems: 3. return to equilibrium • NOTE: the standard approach based on < L ∞ , L 1 > seems to be not effective.
• entropy -the function x → xlogx is appearing; to support this choice we note  Gdansk, December 12, 2013 • The set of "good" density matrices {̺ : S(̺) < ∞} (Wehrl, Rev. Mod. Phys, (1978)) is a meager set only (we assume that the dimension of the underlying Hilbert space is infinite!).
• • Pistone-Sampi: The Orlicz space based on an exponentially growing function cosh − 1 is a "good" space for a description of regular observables!
• An argument, in favour of Orlicz spaces, was provided by Cheng and Kozak (J. Math. Phys;1972). Namely, it seems that the framework within which certain non-linear integral equation of Statistical Mechanics can be studied is that provided by Orlicz spaces.
• mathematical problems associated with Boltzmann's equation; e.g C. • To solve both outlined above problems we propose to replace the pair of Banach spaces < L ∞ (X, Σ, m), L 1 (X, Σ, m) > (1) • by the pair of Orlicz spaces (or equivalent pairs).
• The pair of Orlicz spaces we explicitly use are respectively built on the exponential function (for the description of regular observables) and on an entropic type function (for the corresponding states).
• They form a dual pair (both for classical and quantum systems).

On applications of Orlicz spaces
Applied Math & Math Methods in Physics, Gdansk, December 12, 2013 • This pair has the advantage of being general enough to encompass regular observables, and specific enough for the latter Orlicz space to select states with a well-defined entropy function.
• Moreover for small quantum systems this pair is shown to agree with the classical pairing of bounded linear operators on a Hilbert space, and the trace-class operators.
• The proposed quantization differs between "large systems" and "small systems".
• Consequently, the (Köthe) dual space consist of more regular states. This is a new way of removing "non-physical" states which lead to infinities. Thus a kind of renormalization is proposed. Orlicz spaces (Bennet, Sharpley; Krasnosielsky, Rutickij) • Basic idea: whereas L 1 (m), L 2 (m), L ∞ (m) and the interpolating L p (m) spaces may be regarded as spaces of measurable functions conditioned by the functions t p (1 ≤ p < ∞), the more general category of Orlicz spaces are spaces of measurable functions conditioned by a more general class of convex functions; the so-called Young's functions is said to be a Young's function.
We will assume that Young's functions are equal to 0 for x = 0. • Definition 3. Let Ψ be a Young's function, represented as in (3) as the integral of ψ. Let

A Young's function Φ is said to satisfy
Then the function is called the complementary Young's function of Ψ.
• We note that if the function ψ(w) is continuous and increasing monotonically then φ(v) is a function exactly inverse to ψ(w).

On applications of Orlicz spaces
Applied Math & Math Methods in Physics, Gdansk, December 12, 2013 • Define (another Young's function) • Let L 0 be the space of measurable functions on some σ-finite measure space (X, Σ, µ). We will always assume, that the considered measures are σ-finite.

On applications of Orlicz spaces
Applied Math & Math Methods in Physics, Gdansk, December 12, 2013 • To understand the role of Zygmund spaces the following result will be helpful hold for all p satisfying 1 < p < ∞. Moreover, L exp may be identified with the Banach space dual of LlogL.
• Finally, we will write F 1 ≻ F 2 if and only if F 1 (bx) ≥ F 2 (x) for x ≥ 0 and some b > 0, and we say that the functions F 1 and F 2 are equivalent, • These results lead to Corollary 9. Let (X, Σ, m) be a probability space. Putting Ψ(x) = x log(x + √ 1 + x 2 ) − √ 1 + x 2 + 1 and F (x) = kx log x where k > e is a fixed positive number we obtain: H(f ) is finite provided that f ∈ L Ψ + .
• One has • Theorem 10. Let Φ i , i = 1, 2 be a pair of equivalent Young's function. Proposition 11. Let (Y, Σ, µ) be a σ-finite measure space and L log(L + 1) be the Orlicz space defined by the Young's function x → x log(x + 1), x ≥ 0. Then L log(L + 1) is an equivalent renorming of the Köthe dual of L cosh −1 .
Proposition 12. For finite measure spaces (X , Σ, m) one has Consequently, for the finite measure case, L cosh −1 , L log(L + 1) is an equivalent renorming of L exp , L log L .
• Note also: the functions xlog(x+1) and x log(x+ On applications of Orlicz spaces Applied Math & Math Methods in Physics, Gdansk, December 12, 2013 Regular classical systems • Let {Ω, Σ, ν} be a measure space; ν will be called the reference measure.
The set of densities of all the probability measures equivalent to ν will be called the state space S ν , i.e.
E(f ) ≡ f dν. f ∈ S ν implies that f dν is a probability measure.

On applications of Orlicz spaces
Applied Math & Math Methods in Physics, Gdansk, December 12, 2013 • We define for a stochastic variable u on (Ω, Σ, f dν) • and to have a selection procedure: Definition 14. The set of all random variables on (Ω, Σ, ν) such that for a fixed f ∈ S ν 1.û f is well defined in a neighborhood of the origin 0, 2. the expectation of u is zero, will be denoted by L f ≡ L f (f · ν) and called the set of regular random variables (these conditions imply that all moments are finite!).
• It was proved Theorem 15. (Pistone-Sempi) L f is the closed subspace of the Orlicz space L cosh −1 (f · ν) of zero expectation random variables.
• Note that there is the relation ≻ between the Young's function xln(x + √ 1 + x 2 ) − √ 1 + x 2 + 1 and the entropic function c · xlnx where c is a positive number. Consequently, the condition f ∈ L xln (f · ν) guarantees (for finite measure case) that the continuous entropy is well defined. • the proposed approach is compatible with a rigorous analysis of Boltzmann's equation (infinite measure case).
• Why? • (spatially homogeneous) Boltzmann's equation: where , etc, are velocity distribution functions, with v standing for velocities before collision, and v ′ for velocities after collision. I(g, θ) denotes the differential scattering cross section, dΩ is the solid angle element, and g = |v|.
• The natural Lyapunov functional for this equation is the continuous entropy with opposite sign, i.e.
where f is supposed to be a solution of Boltzmann's equation.

On applications of Orlicz spaces
Applied Math & Math Methods in Physics, Gdansk, December 12, 2013 • The important point to note here is the fact that DiPerna-Lions, Villani showed that the estimates and where f 1 f 2 , are sufficient to build a mathematical theory of weak solutions (faithful citation; x stands for coordinate -is appearing in non-spatially homogeneous Boltzmann equation).
• Furthermore, Villani announced that for particular cross sections (collision kernels in Villani's terminology) weak solutions of Boltzmann equation are in L log(L + 1).

On applications of Orlicz spaces
Applied Math & Math Methods in Physics, Gdansk, December 12, 2013 • Also, H + (f ) is nicely defined, provided that f ∈ L log(L + 1).
• Consequently, the scheme for classical statistical mechanics based on the two distinguished Orlicz spaces L cosh −1 , L log(L + 1) does work. However, the basic theory for Nature is Quantum Mechanics. Therefore the question of a quantization of the given approach must be considered.
• However, there is a problem: • " To the malicious delight of many mathematicians the initial impression that type III is the rule for infinite systems has panned out with the passage of time. Types I and II turns out to be peripheral possibilities" Non-commutative Orlicz spaces • Let A be a von Neumann algebra acting on a Hilbert space H with normal faithful semifinite weight ϕ • Generate a bigger algebra M on L 2 (IR, H), so called cross product M = A × σ IR • M is generated by π(x), x ∈ A and λ s , s ∈ IR.
• a is affiliated with M (denoted aηM) if u and all the spectral projections of |a| belong to M.
• Denote by M the set of all τ -measurable operators.
• Haagerup's approach to non-commutative integration. • g(s(a)) gives a nice norm on the matrix algebra, where s(a) is a vector formed from singular values of a while g stands for a symmetric gauge functional. a stands for a n × n matrix.
• The function t → µ t (f ) will generally be denoted by µ(f ).
• A function norm ρ on L 0 (0, ∞) is defined to be a mapping ρ : • Such a ρ may be extended to all of L 0 by setting ρ(f ) = ρ(|f |).
If now L ρ (0, ∞) turns out to be a Banach space when equipped with the norm ρ(·), we refer to it as a Banach Function space. • If ρ(f ) ≤ lim inf n (f n ) whenever (f n ) ⊂ L 0 converges almost everywhere to f ∈ L 0 , we say that ρ has the Fatou Property.
• If this implication only holds for (f n ) ∪ {f } ⊂ L ρ , we say that ρ is lower semi-continuous.
• When M = L ∞ (X, m) and τ (f ) = f dm one gets • so, µ t (f ) is exactly the classical non-increasing rearrangement f * (t).

On applications of Orlicz spaces
Applied Math & Math Methods in Physics, Gdansk, December 12, 2013 • Dodds, Dodds and de Pagter formally defined the noncommutative space and showed that if ρ is lower semicontinuous and L ρ (0, ∞) rearrangement-invariant, L ρ ( M) is a Banach space when equipped with the norm f ρ = ρ(µ(f )).
• For any Young's function Φ, the Orlicz space L Φ (0, ∞) is known to be a rearrangement invariant Banach Function space with the norm having the Fatou Property.
• Thus taking ρ to be · Φ , the very general framework of Dodds, Dodds and de Pagter presents us with an alternative approach to realizing noncommutative Orlicz spaces.
• Definition 17. The noncommutative statistical model consists of a quantum measure space (M, τ ), "quantum densities with respect to τ " in the form of M +,1 * ,0 , and the set of τ -measurable operators M.
(Notice that the requirement that 0 ∈ D( µ g x (t)) 0 , presupposes that the transform µ g x (t) is well-defined in a neighborhood of the origin.)

On applications of Orlicz spaces
Applied Math & Math Methods in Physics, Gdansk, December 12, 2013 • We remind that above and in the sequel µ(g) (µ(x)) stands for the • To give a non-commutative generalization of Pistone-Sempi theorem we need a generalization of Dodds, Dodds, de Pagter approach i.e. that one which just was presented • Definition 19. Let x ∈ L 1 + (M, τ ) and let ρ be a Banach function norm on L 0 ((0, ∞), µ t (x)dt). In the spirit of Dodds, Dodds, de Pagter, we then formally define the weighted noncommutative Banach function space L ρ x ( M) to be the collection of all f ∈ M for which µ(f ) belongs to L ρ ((0, ∞), µ t (x)dt). For any such f we write f ρ = ρ(µ(f )).
• Remark 20. Comparing commutative and non-commutative regular statistical models, we note that µ t (x) (the Lebesgue measure dt) in Definition 19 stands for f (dν, respectively).

On applications of Orlicz spaces
Applied Math & Math Methods in Physics, Gdansk, December 12, 2013 • The mentioned generalization of Dodds, Dodds, de Pagter approach is contained in: Theorem 21. Let x ∈ L 1 + (M, τ ). Let ρ be a rearrangement-invariant Banach function norm on L 0 ((0, ∞), µ t (x)dt) which satisfies the Fatou property, ρ(χ E ) < ∞ and E f dµ ≤ C E ρ(f ) for E : µ(E) < ∞. Then L ρ x ( M) is a linear space and · ρ a norm. Equipped with the norm · ρ , L ρ x ( M) is a Banach space which injects continuously into M.
• and the generalization of Pistone-Sempi is given by • To show that statistics and thermodynamics can be well established for noncommutative regular statistical systems, we note that Proposition 23. Let M be a semifinite von Neumann algebra with an fns trace τ and let f ∈ L 1 ∩ L log(L + 1)( M), f ≥ 0. Then τ (f log(f + ǫ)) is well defined for any ǫ > 0. Moreover is bounded above, and if in addition f ∈ L 1/2 (equivalently f 1/2 ∈ L 1 ), it is also bounded from below. Thus τ (f log f ) is bounded below on a dense subset of the positive cone of L log(L + 1).
• Consequently, if the "state" is taken from the "good" noncommutative Orlicz space, then the entropy function is well defined.

On applications of Orlicz spaces
Applied Math & Math Methods in Physics, Gdansk, December 12, 2013 • Analogous to the commutative case, we get the following conclusion: • Corollary 25. Either of the pairs L cosh −1 , Llog(L + 1) or L exp , L log L provides an elegant rigorous framework for the description of noncommutative regular statistical systems, where now the Orlicz (and Zygmund) spaces are noncommutative.