# Are there ”small cardinal models” of a Banach space, whose dual space is in
Stegall‘s class, but it is not $weak^{*}$-fragmentable, or large cardinals are
a must?

Svilen I. Popov  Svilen I. Popov, Charles University, Faculty of Mathematics
and Physics, Sokolovská 83, 186 75 Praha, Czech Republic
<EMAIL_ADDRESS>
(Date: December 2023)

On Consistency and Independence of models beyond ZFC implying strictness of
the inclusions of classes of Banach spaces which are in Stegall‘s class $S$,
but not having $weak^{*}$ \- fragmentable dual balls.

## Abstract

It is well-known that if $Y$ is a Banach space the $weak^{*}$-fragmentability
of its dual space by some metric $\rho$ implies that $Y^{*}$ belongs to the
Stegall class - the former for shortly $\mathcal{W^{*}}F$, being the latter
$\mathcal{S}$ and hence $Y$ is weak Asplund - call it $\mathcal{WA}$ . It has
been proved by O. Kalenda and Kunen that existence of a measurable cardinal
implies (it is consistent) that, for instance in the construction of Kalenda
Compacts - this space is in the Stegall‘s class iff both inclusions of classes
are strictly proper. The same authors made following question, is there a
model of ZFC, in which the inclusion of $\mathcal{WA}$ in $\mathcal{S}$
actually is equality. Obviously, because of their result the existence of
large cardinals ( supercompacts, strongly-compacts, strong cardinals, huge
cardinals, Vopěnka principle) would be a models of the proper inclusion of the
above - mentioned classes, being with more consistency power even, see Thomas
Jech [TJ03] and Saharon Shelah [SSH17].

## Introduction

Let us start our journey by the following facts:

(*) Every continuous function from a Baire metric space $B$ into $A$ is
constant on some non-empty open subset of $B$.

A double arrow space constructed by using $A$ say it $K_{A}$, see [KK05] gives
us an example of a Stegall space, if A satisfies (*). And $K_{A}$ is
weak∗-fragmentable by some metric, i.e, $B_{C^{*}(K_{A})}(0,1)$ is
fragmentable in weak∗ topology by some metric iff $A$ is at most countable. In
particular, the Banach space ${C(K_{A})}$ is withness of the properness of the
$weak^{*}$-fragmentable spaces by some metric in Stegall class iff $A$ is
uncountable. More precisely, by [KK05] the above construction is divided by
two logical approximations:

Theorem 1

  1. (1)

$(K_{A})$ is in the class $weak^{*}$-fragmentable spaces by some metric iff
$A$ is at most countable.

  2. (2)

$(K_{A})$ lies in the Stegall‘s class iff $A\leq\aleph_{1}$. (Because
satisfies (*)).

These approximations in [KK05] relies on set-theoretic assumptions weaker in
consistency to the following model:

Definition ($\mathcal{A!}$) Let us assume that:

  1. (1)

$MA(\aleph_{1})+2^{\aleph_{0}}=\aleph_{3};$

  2. (2)

The Nonstationary ideal on $\aleph_{2}$ is precipitous.

  3. (3)

$\aleph_{1}$ is not measurable cardinal in any transitive model of ZFC,
containing all ordinals.

Lemma 1.

($!\mathcal{A}$) is equiconsistent with the existence of a measurable cardinal
of Mitchell order 2.

###### Proof.

In [KK05] it has been shown that the existence of a measurable cardinal is
equiconsistent to ($\mathcal{A!}$), and see Th. Jech [TJ03, Th. 38.5] for the
iff directions of the equiconsitency of the second point in (!$\mathcal{A!}$)
and the existence of the measurable cardinal of Mitchel order $2$.

Now, after this humble modification let us start our approximation work,
namely let us introduce the following principle ($\mathcal{B}!$):

It is well-known that if $Y$ is a Banach space the $weak^{*}$-fragmentability,
of its dual space by some metric $\rho$ i.e ($W^{*}F$-spaces) implies that
$Y^{*}$ belongs to the Stegall class $S$ and hence $Y$ is weak Asplund - call
it $\mathcal{WA}.$

In the traditional general topology setting all researchers are suggesting
models either based on or including $MA\ \ +\ \ \neg\ CH$ $+$ $ZFC$ as an
opposition to the usual very rough model of the Gödel Constructibility
Universum Axiom, i.e ($V=L$) $+$ $ZF$.Therefore there is no big surprise in
the need of effective set-theoretic approximations, being improvement of the
existing models.

For instance, the above results in the Topology of the Functional Analysis
have been constructed and established in the paper of Kalenda and Kunen
[KK05].

What we suggest is to improve the models of small cardinals $MA\ \ +\ \ \neg\
CH$ $+$ $ZFC$ and ($V=L$) $+$ $ZF$, keeping the size of the main ingradient of
the Kalenda Compact construction obtained by the modification of the famous
Solovay-Tannenmbaum model of ZFC, used in [KK05]. So, it turns out that subtle
set-theoretic topological principles and constructions stand behind the scene.
For instance, there is a hidden connection with the famous Whitehead Problem.
So, we will establish the new kind of a principle, solving, partly the Tall-
Moore problem as well, combining this with the considered of the current paper
Kalenda problem i.e:

Kalenda Problem 1.:

Under which model of ZFC there exists a Banach space, whose dual space is
properly in Stegall‘s class, are there models in which there is no such a
space and what about weak- ∗ fragmentable spaces.

We partly answer the question from above reducing it to the following
statement:

question ”Are there $Q$-sets $A_{1},A_{2}$ whose cardinality are $\aleph_{1},$
$\aleph_{2}$ respectively, can we replace the requirement ”strictly less than
the least inaccessible cardinal in $L$” by ”$\exists\ S\subset
E^{\aleph_{2}}_{\aleph_{0}},$ such that $S$ is stationary, non-reflecting”.

Last sentence is where we actually enter into the game, so we will closely
inspect the last question. We will spend some words on the consistency of the
model in the paper [KK05] as well. See Thomas Jech and Saharon Shelah, i.e
[TJ03] and [SSh17], for the question, for instance, about the role of large
cardinals and some knowledge about them, topics treated in [KK05].

## Basic Model and Independence Results

Now, let us again return to our main problem  Kalenda Problem 1.:

Firstly, we show the independence of the weak diamond and the existence of a
$Q$ set from the following statement $\mathcal{K}$:

Kalenda Compacts $K_{A}$ from the construction [KK05] are properly in
Stegall‘s class

Let us remind that $K_{A}$ if it is the corresponding Kalenda Compact,
constructed in the paper and endowed with the topology of the usual double
arrow space of the consideration with the aid of $A$ \- a subset of the
interval $(0,1).$

Proposition The weak diamond at $\aleph_{1}$ is independent from
$\mathcal{K}+ZFC$, in particular the existence of a $Q$ set is independent
from $\mathcal{K}+ZFC.$

###### Proof.

By [NP92] $\mathcal{K}+ZFC$ holds under $V=L$, in particular the principle
suggested there implies $CH.$ On the other hand, under $CH$ does not exist a
proper $Q$-set, i.e in particular, uncountable strongly measure zero set,

On the other hand, as we show in the main result a $Q$-set exists under
special kind of a uniformization, in particular this implies
$2^{\omega}=2^{\omega_{1}},$ i.e a contradiction with the weak diamond at
$\aleph_{1}.$ By [KK05] a model of existence of a $Q$-set implies that the
space $K_{A}$ belongs to the Stegall‘s class by taking $A$ to be the
corresponding $Q$-set - being uncountable immediatedly gives us the non-weak ∗
-fragmentability of the Kalenda compact.

∎

###  0.1. Model of Existence of $Q$-set

Assume MA ($\aleph_{0}$ \- centered) $\ \ +\ \ \neg\ CH$ $+$ $ZFC$ more
generally, we can replace the requirement ”strictly less than the least
inaccessible cardinal in $L$” by  ”$\exists\ S\subset
E^{\aleph_{2}}_{\aleph_{0}},$ such that $S$ is stationary, non-reflecting” .
The model we suggest is based on the uniformization properties of MA
($\kappa$-centered), $\forall\ \aleph_{0}\leq\kappa<2^{\aleph_{0}}$ $+$
$2^{\aleph_{0}}=\aleph_{3}.$

Now, in order to reverse the direction and to make the implication ”if and
only if” for the statement:

”The double arrow space $K_{A}$ is in Stegall‘s class, not w∗-fragmentable by
any metric if and only if $A$ is a $Q$-set, whose cardinality is exactly
$\aleph_{1}$”.

Let us assume that $A$ has cardinality more or equal to $\aleph_{2}$. Then
since as in [KK05] there is a precipituous ideal on $\aleph_{2}$. It turns out
that this condition as it well-known for the case $\aleph_{1}$, see exactly
[NP92], the game and the construction there, implies the result, in
particular, we obtain form of generalization of that result - depending
strongly on the choosen scattered compact with the defined property
$\mathcal{N}$, giving some insight to the weak-fragmentability and, in
specific, to the Eberlein Compacta.

Kenderov [PW01], firstly establishes a construction in ZFC about conditions
about weak Asplundness of a space, not weak∗-fragmentable - we will make a
note on the result below in the examples - of course any constructions about
non-existence of a weak asplund space, not weak∗ fragmentable should rely on
additional set-theoretic assumption.

Now, after this humble clarification and modification of $\mathcal{A!}$ let us
start our approximation work, namely let us introduce the following principle:

Definition $\mathcal{B!}$

  1. (1)

$2^{\aleph_{0}}=\aleph_{3}$ and let the weak square at
$\square_{\aleph_{1},\aleph_{1}}$ hold.

  2. (2)

Now, let us assume the following principle call it (***)

    1. (a)

let if $Unif\\{\aleph_{1},\aleph_{0},\aleph_{0}\\}$ holds then, precisely
$\exists\ \ S\ \notin Id-Unif\
\\{\aleph_{1},\aleph_{0},\aleph_{0}\\}:=I^{\aleph_{1}}$ and non-reflecting.
Differently saying let on $S$ there be at least a countable colouring
uniformization on the ladder system, given by $S$, i.e making it non-
reflecting.

    2. (b)

And in the case of $\aleph_{2}$, let $S$ be the set given by the square above,
moreover let if $Unif\\{\aleph_{2},\kappa,\mu\\}$ holds for $\kappa,\mu$
witnesses of the colours and the ladder systems say ($\omega$-chromatic and
$\aleph_{1}$-chromatic) then let $S\ \notin Id-
Unif\\{\aleph_{2},\kappa,\mu\\}:=\mathcal{I}^{\aleph_{2}}$ \- again let the
ladder is the one consistently existing from the conditions at the beginning,
making $S$ to be non-reflecting, see Shelah [SSh17].

Theorem

$\mathcal{B!}$ is not consistent with the assertion:

$\aleph_{2}$ is a measurable cardinal in a transitive model of $ZFC$,
containing the ordinals

The model of Stegall‘s Banach space, not weak ∗-fragmentable.

Before we continue, let us clarify the following notations in the model given
by $ZFC+\mathcal{B}!.$ Most Importantly, we denote by
$S_{i},\notin\mathcal{I}^{\aleph_{i}},$ where $i\in\\{1,2\\},$ stands for the
two stationary sets given by the suggested model whereas the corresponding
domain ideals of the uniformizations shall be denoted as usual by
$\mathcal{I}^{\aleph_{i}}$, $i\in\\{1,2\\}$. Therefore the next result shall
be formulated as the following proposition, namely:

Proposition (Under $\mathcal{B}!$)

By using the notations above let us assume that $\aleph_{i}$ is not measurable
in any transitive model containing all ordinals and, moreover, they do not
carry any precipitous ideals. Then if the size of $A$ is less or equal to
$\aleph_{2}$ then $K_{A}$ from [KK05] is in the Stegall’s class.

###### Proof.

It is well-known that the uniformization with $\aleph_{0}$ ($\aleph_{1}$) -
colourings on the sets $S_{1},\ S_{2}$ provide us normal, Moore spaces, first
countable, separable, whose density is ( $\aleph_{1},$), not collectionwise
Haussdorff $\aleph_{1},$ ($\aleph_{2}$), this implies the existence of $Q,$
sets, whose size is $\aleph_{1}$ and $\aleph_{2},$ respectively. By [KK05] the
corresponding $K_{A}$ belongs to Stegall’s class, since for all metrizable
Baire spaces, there exists a continuous map from them to $A,$ constant on a
nonempty open subset.  ∎

###  0.2. Independence of the Model, regarding $K_{A}$

We begin the subsection with the following theorem:

Theorem

  1. (1)

PFA hold;

  2. (2)

Let $\aleph_{1}$ is not measurable in any transitive models of ZFC, containing
all ordinals

  3. (3)

Let the ideal $\mathcal{I}^{\aleph_{1}},$ contains all non-reflecting sets on
$\aleph_{1},$ see [SSh17].

  4. (4)

Last but not least, let $\aleph_{2}$ carries a precipitous ideal

  5. (5)

let us well suppose the existence of a measurable cardinal, for instance
$\aleph_{2}.$

Then $K_{A}$ is in Stegall‘s class iff it is weak ∗ \- fragmentable.

###### Proof.

Regarding the first uncountable cardinal in $V,$ $\aleph_{1}$ we finished the
proof, since there are two possible situations:

  1. (1)

Either there exists $a\subset\omega$ such that $\aleph_{1}^{L[a]}$ is
uncountable. ([TJ03, TH. 25.38])

  2. (2)

Or a Tree Uniformization holds on a Special tree on $\aleph_{1},$ (Aronszajn
tree).

For the first part (item) of the enumeration, it is well-known that it is
independent of ZFC, see [NP92], assuming it under ($V=L$), for instance. As
well, the first item implies $CH,$ however this is a contradiction with the
assumption, since in $V$ we assumed that $\aleph_{2}=2^{\aleph_{0}}.$

On the other hand the second item is impossible in our model as well, there is
no such a uniformization with the required properties. Regarding $\aleph_{2}$
by the model we assumed it follows that $\aleph_{2}$ is weakly compact in $L,$
henceforth $\diamondsuit(\aleph_{2})$ holds and as it is well-known there are
no $Q$-sets with the size of weakly compact cardinal. Regarding the
possibilities of $V=L$ \- like principles we claim that there is no any
analytic or co-analytic subset of the real line up to the third level of the
projective hierarchy, (the reason is the existence of a precipitous ideal on
$\aleph_{2}$ and the assumption of a measurable cardinal existence), see
Magidor proof in [TJ03, Th. 32.16].

∎

###  0.3. Two suggestions of models.

Proposition 3.

Proposition 3.

  1. (1)

Let us suppose that on $\aleph_{3}$ there exist a precipitous ideal;

  2. (2)

Let $\aleph_{1}$ is not measurable in any transitive models of ZFC, containing
all ordinals, and let us assume that on $\aleph_{1}$ there are no precipitous
ideal.

  3. (3)

Let $\mathcal{I}^{\aleph_{1}}$ be the ideal corresponding to all
uniformization on $\aleph_{1}$ and contains all non-reflecting sets on
$\aleph_{1},$ see [SSh17].

  4. (4)

let as well on $\mathcal{B}!$ hold with the modification that the real line
has a size $\aleph_{4}$

  5. (5)

let us well suppose the existence of a measurable cardinal.

$K_{A}$ that it is in Stegall‘s class if and only if $A$ has size less or
equal to $\aleph_{2}$.

Proposition 4.

  1. (1)

Let $\aleph_{1}$ be not measurable in any transitive models of ZFC, containing
all ordinals, and let us assume that on $\aleph_{1}$ there is no precipitous
ideal.

  2. (2)

let us suppose that only the first item of $\mathcal{B}!$ hold, and let either
$\aleph_{2}$ be measurable cardinal or $S_{2}\in\mathcal{I}^{\aleph_{2}}$) be
non-reflecting, stationary.

  3. (3)

let as well on $\mathcal{B}!$ hold with the modification that the real line
has a size $\aleph_{4}$

  4. (4)

let us well suppose the existence of a measurable cardinal.

$K_{A}$ that it is in Stegall‘s class if and only if $A$ has size less or
equal to $\aleph_{2}$.

Let on $\aleph_{1}$ there is no precipitous ideal and as well, it is not
measurable in any transitive model of ZFC containing the ordinals, , then
$K_{A}$ is in Stegall‘s class iff $A$ has size at most $\aleph_{1}.$

###  0.4. Applications of Stegall’s class and precipitous at $\aleph_{1}$

By [NP92] we remind that property $\mathcal{N}$ stands for the assertion (!*)
where

Definition: (!*) [NP92]

(!*) The point of continuity of every continuous function from a Baire metric
space $B$ into $(C(K)\phi_{p})$ is dense in $B$. The symbol $(C(K)\phi_{p})$
stands for the continuous functions over $K,$ endowed with the point-wise
convergence topology. [NP92]

The property (!*) generalizes the notion of the fragmentability and, in
particular, it is implied by the special case of Eberlein compacta, i.e weak
fragmentable by the norm.

Remark 1. (NP92)

By [NP92] (under CH) any scattered compact $K$ with property $\mathcal{N}$
admitting stronger topology than the euclidean one such that in it there is a
uncountable locally compact, Hausdorff subset of the unit interval $Y$, such
that the one-point compactification of $Y$ is $K.$ Has actually hereditary
Lindeloff space of continuous functions over $K$ endowed with point-wise
topology on it.

Remark 2. [NP92]

By [NP92] the existence of precipitous ideal on $\aleph_{1}$ implies that if
$(C(K),\phi_{p})$ is hereditary Lindeloff and $K$ being a scattered compact,
then:

  1. (1)

$K$ has the property (!*) iff $K$ is countable.

Proposition

(!*) implies that, for general compact spaces $K,$ the property $*$ holds.

Corollary

Let us consider model of CH $+$ $\exists$ precipitous ideal on $\aleph_{1}$.

Then there is no $a\subset\omega$ such that $\aleph_{1}^{L[a]}$ is
uncountable.

Proof

Assume for contradiction the opposite, then by [NP92] $K$ has (!*), because we
assumed by contradiction (**), see [TJ03], then by Remark 1 and 2. [NP92] $K$
should be uncountable - contradiction.

Definition

Let $K$ be a scattered compact, we call Namioka‘s class Banach spaces - the
space of the continuous functions over $Y,$ i.e $C(Y)$ if $K$ has the
Namioka‘s Property $!^{*}$ and $K$ is the one-point compactification of
$Y\subset[0,1]$ of $(Y,\phi)$ where $(Y,\phi)$ is a locally compact topology
strictly stronger than the euclidean one.

Corollary

Any $C(Y)$ is in Namioka‘s class satisfies $(*),$ with respect to all
completely regular Baire spaces, see [KS99], for details, i.e is in Stegall‘s
class S.

Corollary (Under CH $+$ $\exists$ precipitous ideal on $\aleph_{1}$).

Any $C(Y)$ in Namioka‘s class is weak ∗ fragmentable partly answering to the
Kalenda‘s question if the class of Stegall‘s compact spaces coincide to the
weak ∗ fragmentable compacts under the model:  CH $+$ $\exists$ precipitous
ideal on $\aleph_{1}$.

###  0.5. Consistency Power

Furthermore, we show the consistency of our model with the assertion that the
precipitous ideal on $\aleph_{2}$ is exactly the ideal of the non-stationary
sets $\mathcal{I}^{\aleph_{2}}_{\mathbf{NS}}$, by Shelah we know that the
uniformizations at any cardinal are not valid on a normal ideal containing
$\mathcal{I}^{\kappa}_{\mathbf{NS}}.$ Moreover the ideal $\\{\kappa\in S|Id-
Unif\\}$ should be proper.

Now, since, as it is well-known by Magidor - see Jech [TJ03] the power of the
consistency of this model is exactly as such as of the existence a cardinal
$\kappa$ of Mitchel order $2,$ namely $\kappa$ is a measurable cardinal, which
stays measurable cardinal of in its Ultrapower, taken by the non-principal
normal measure on $\kappa,$ say it $[U]$ witness that $\kappa$ is a measurable
cardinal.

Requirement

Since by Shelah [SSH17] the dual ideal of the domain of the uniformizations at
any cardinal is a normal ideal $I_{U}$ containing
$\mathcal{I}^{\kappa}_{\mathbf{NS}}.$ In order to avoid difficulties of a set-
theoretic character we will assume that $I_{U}$ strictly contains
$\mathcal{I}^{\kappa}_{\mathbf{NS}},$ more precisely let it be the normal
closure of $\mathcal{I}^{\kappa}_{\mathbf{NS}}.$ This technical requirement is
based on the fact that $I_{U}$ should be a normal ideal, whereas it is an open
problem in Set Theory if $\mathcal{I}^{\kappa}_{\mathbf{NS}}$ is a normal
ideal, when it is precipitous at the same time.

Now, assuming ($\mathcal{B}!$), there exists a $Q$-set with size $\aleph_{1}$
if and only if $Unif\\{\aleph_{1},\aleph_{0},\aleph_{0}\\}$ holds.

($\mathcal{B}!$) is strictly weaker than the model

$2^{\aleph_{0}}=\aleph_{3}+MA(\sigma-centered)+MA(\aleph_{1}-centered)$

and it is implied by it.

Proof

See, Kunen [KKJV14] about how $MA(\sigma-centered)$ implies the first item of
($\mathcal{B}!$), now see [TJ03] for the second item of ($\mathcal{B}!$) and
it is well-know the opposite, see Kunen for a counter example of
($\mathcal{B}!$) and why the requirements are important in the principle, as
well, see [TJ03] the main result about the equivalence of the second item and
the main condition of ($\mathcal{B}!$), with for instance, the non-weak-
compactness of $\aleph_{2}$ if you wish, dear reader, with the inconsistency
with the very PFA regarding $\aleph_{2}$, for instance, [TJ03] and [SSh17].

Remark

Actually, we can assume that $\mathcal{B}!$ holds for any of the two
cardinals, more generally, we can assume it over any cardinals (we can
increase the size of the real line, as well, i.e to assume that the continuum
is any cardinal). Let us point out that the first item of ($\mathcal{B}!$)
implies both that $\aleph_{2}$ is not weakly compact, as well we know that
this notion is preserved in $V$ and in $L$ and the first item of
$\mathcal{B}!$ is equivalent to existence of a tree on $\aleph_{2}$ which is
Moore space (first countable and separable, of course, nonmetrizable - never
it can be). Now, by [ShJT01] Shelah we know that we do not need a two-
chromatic uniformization for converting a $T_{1}$ topological space in
$T_{4}$, monochromatic uniformization is pretty enough.

As well for completeness, let us mention that actually two-chromatic
uniformization is equivalent to $\omega$-chromatic uniformization. So, in
particular, we can establish the following well-known result: If we restrict
the Principle $\mathcal{B}!$ only regarding the behaviour of $\omega$, the
exponent of $\omega$ and $\omega_{1}$ and in addition, we assume
$2^{\aleph_{0}}=2^{\aleph_{1}}$ then this is equivalent to existence of a non-
projective, $R$-projective module with size exactly $\aleph_{1}$ over a
commutative non-perfect ring $R$ (in general, the existence of such a module
is strictly stronger than the normalization of a topological space), so the
requirement of the normalization is a model not provable in ZFC, as well, and
since by Namioka we know that in $V=L$ there exists a Stegall space, not
weak∗-fragmentable, as well by the previous Theorem we deduce the same in the
model of Principle $\mathcal{B}!$ only regarding the behaviour of $\omega$,
the exponent of $\omega$ and $\omega_{1}$ and in addition, if we assume
$2^{\aleph_{0}}=2^{\aleph_{1}}$, however this is strongly opposite to the
solution of the Dual Baer test $R$-module problem,see e.g. [JT20], since the
principle of Namioka [PW01], established below implies CH and this implies the
negation of the usual uniformization of $\aleph_{1}$ that we assume, which is
the famous weak diamond at $\aleph_{1},$ discovered by Shelah and Devlin
[KDSSh78] and we know very well, that in this case a module with size
$\aleph_{1}$ with $proj.dim=1$ cannot be $R$-projective module over any ring,
not the case implicitly said and as it is well - known by the algebraic
persons, this is the situation, equivalent to the existence of a $Q$-set with
size $\aleph_{1}$ over the real line and, for instance, assuming that
Principle $\mathcal{B}!$ holds, i.e we have a model with existence of a
Stegall Banach space, whose dual is not weak∗-fragmentable, but with positive
solution of the Dual Baer test $R$-module problem, see Trlifaj [JT20].

Theorem

Let $\aleph_{2}$ be a weakly compact cardinal and henceforth inaccessible in
$L$. Then in particular, the negation of the weakly Kurepa Hypothesis
($\neg$wKH) also $2^{\aleph_{0}}=\aleph_{2}$ imply:

  1. (1)

$2^{\aleph}_{0}=2^{\aleph}_{1}$

  2. (2)

$\diamondsuit(\aleph_{2}).$

Corollary

Let the previous theorem hold and $\mathcal{B}$ item 1.) then if there is a
coseparable abelian group of size $\aleph_{1}$ then there is a $Q$-set of size
$\aleph_{1}$ and there is no $Q$ \- sets of size $\aleph_{2}.$

Proof

Henceforth there is no $Q$ \- set of size $\aleph_{2}$, since in a presence of
$\diamondsuit(\aleph_{2})$ any $\aleph_{2}$-c.c normal Moore topological space
is metrizable and a $Q$ \- set of size $\aleph_{2}$ is equivalent to an
existence of a normal Moore space with density $\aleph_{1}$, which is non-
metrizable, which is not the case when $\diamondsuit(\aleph_{2})$ holds.
Moreover, then any Whitehead group of a size $\aleph_{2}$ is free.

Corollary Let the previous theorem and $\mathcal{B}$ item 1.) hold then:

  1. (1)

There is a right $R$ \- module admitting a minimal right generating set of
size $\aleph_{1}$, whose right-projective dimension is exactly 1 ( Mitag-
Leffler flat), but being right $R$-projective over $R$ being a right
hereditary ring.

Moreover we have as well:

  2. (2)

Any right $R$ \- module admitting a minimal right generating set of size
$\aleph_{2}$, in whose filtration (from the definition, of
$\aleph_{2}$-strongly projectiveness, see i.e [RGJT06] do not participate any
of the submodules given by item 1.) then it is indeed right-projective, i.e
modulus some modules that should not participate in the filtration of the
right $R$ \- module from above any right $R$ \- module with a generation set
of size $\aleph_{2}$ is indeed right projective over the ring, for the
definitions and the constructions see, i.e Shelah-Trlifaj [ShJT01].

  1. !!!

In particular, there exists a Whitehead group of size $\aleph_{1}$ non-free,
and any Whitehead group of size $\aleph_{2}$, whose set of submodules,
admiting $\aleph_{1}$ \- generation and being Whitehead groups does not
involve the modules of the kind ”Whitehead group of size $\aleph_{1},$ but
non-free”, are free.

Remark

  1. !!

So, we see that the requirements from $\mathcal{B}$ item 2.) are essential,
since $\mathcal{B}$ item 1.) exists in ZFC without the requirement of the
uniformization behaviour. Moreover the previous theorem implies the negation
of $\mathcal{B}$ item 2.).

  1. !!!!

Now, in the context of O. Kalenda paper the requirement that $\aleph_{2}$ is
not inaccessible in $L$ is essential by the previous theorem, actually what we
need is our principle $\mathcal{B}$ being not in a contradiction with the
assertion $\aleph_{2}$ is inaccessible in $L$. Let us point out that in that
case a special $\aleph_{2}$-Aronzajn tree may exist, although $\aleph_{2}$ is
inaccessible in $L$, however making $\aleph_{2}$ a weakly compact is actually
what by O. Kalenda has been avoided and it is essential to keep in mind that
exactly the assumption that $\aleph_{2}$ is a weakly compact cardinal, being
equal to the Cantor tree, i.e $2^{\aleph}_{0}=\aleph_{2}$ provides us
$\diamondsuit(E^{\aleph_{2}}_{\aleph_{0}})$, which automatically ”kills” any
kind of uniformizations cofinal at $\omega.$ In particular, by the corrollary
above $\diamondsuit(\aleph_{2})$ kills the $Q$-sets of size $\aleph_{2},$
since the essential new $Q$-sets, i.edifferent from $Q$-set of size
$\aleph_{1}$ -which are equivalent to normal, separable, Moore space, non-
metrizable spaces, should require metrization of the space, i.e in a presence
$\diamondsuit(\aleph_{2})$ any $\aleph_{1}$ dense, but strictly not separable,
Moore, normal space, not containing a normal separable non-metrizable subspace
are actually metrizable, so in the case of a dense number of a Moore space
equal to exactly equal to $\aleph_{1}$ $\diamondsuit(\aleph_{2})$ allows you
to make the separability reduction.

Let us recall the following theorem:

Theorem A topological space $Y$ is metrizable if and only if it is a Moore
space, which is collectionwise normal.

Lemma Separability Reduction 1.)

Let $\diamondsuit(\aleph_{2})$ and $\neg\ CH$ hold, then for the locally
compact T$4$ Moore space $Y,$ whose density number is $\aleph_{1}$ and all of
its open subsets are normal spaces [Engelking] (and it is heriditarily Moore)
as well then TFAE:

  1. (1)

$Y$ is metrizable.

  2. (2)

$Y$ does not contain any separable subspace, which have an uncountable closed
discrete subset, see [Engelking].

  3. (3)

$Y$ any $\leq\aleph_{1}$-collectionwise Hausdorff subspace is actually
$\aleph_{2}$-collectionwise Hausdorff.

In the condition of the theorem above we obtain the following result assuming
the following theorem:

Lemma Separability Reduction 2.

Let $\aleph_{2}$ be a weakly compact cardinal and henceforth inaccessible in
$L$, let as well, $2^{\aleph_{0}}=2^{\aleph_{1}}=\aleph_{2}$ hold. Then any
Topological space $Y$ with a dense number $\aleph_{1}$, which is
$<\aleph_{2}$-collectionwise Hausdorff is exactly $\aleph_{2}$-collectionwise
Hausdorff.

Remark The two lemmas above give us a test how to see if a T$6$ heriditarily
Moore space $Y$ whose density number is $\aleph_{1}$ is metrizable, it
suffices to see if all of its separable normal subspaces are metrizable. In
particular, if $\diamondsuit(\aleph_{2})$ and $2^{\aleph_{0}}<2^{\aleph_{1}}$
hold then all normal Moore spaces, whose density number is $\aleph_{1}$ are
metrizable. So, we find out that in terms of normality the lemma gives us a
comparison with the following, well-known but pretty deep results from the
Topology of the Functional Analysis:

Lemma Characterization

(ZFC) For a Banach space $X$ TFAE:

  1. (1)

$X$ is Asplund.

  2. (2)

All of the unit balls of the separable closed subspaces of $X$ are Asplund
sets in the weak topology.

  3. (3)

All of the dual (continuous) spaces of the separable closed subspaces of $X$
are normal in the weak topology of corresponding dual space.

  4. (4)

All of the dual unit balls of the separable closed subspaces of $X$ are weak∗
fragmented by the dual norms.

The items 2.) and 4.) are in the fashion of O. Kalenda, whereas item 3.) is in
the Ribarska style.

So, in particular, assuming the conditions of the lemma separability reduction
we obtain the following result:

Lemma Characterization

Let $\neg\ CH\ +\ \diamondsuit(\aleph_{2})+\mathcal{B}_{1}$ hold, where
$\mathcal{B}_{1}$ stands for the first item of the principle $\mathcal{B}$
then assuming that $\omega_{1}$ is not a measurable cardinal in any transitive
model of ZFC:

  1. (1)

Kalenda compacts $K_{A}$ are in Stegall class, not weak∗ fragmentable by any
metric for $A$, being the $Q$-set, provided by the model above.

  2. (2)

$\neg\ CH\ +\ \diamondsuit(\aleph_{2})+\mathcal{B}_{1}$ or (PFA/MM) is
consistent with existence of many measurable cardinals ( enough ), so let us
assume $\neg\ CH\ +\ \diamondsuit(\aleph_{2})+\mathcal{B}_{1}$ and PMEA then
in this model:

Kalenda Compacts $K_{A}$ are in Stegall‘s class if and only if $A$ is of
cardinality strictly less than $\aleph_{2}$ and Kalenda Compacts $K_{A}$ are
not weak Asplund for any $A$ of cardinality $\aleph_{2}.$

So, in this way, we deduce that the condition of the theorem in the [KK05]
paper, in specific in Kalenda paper alone are important, i.e the statement
where we deduce that $\omega_{2}$ is in not innaccessible in $L$ as well
$\omega_{1}$ of course, since the separability lemma 2. needs this assumption
and with combination with the theorem gives us this characterization.

Remark PMEA stands for the following assertion:

For any cardinal $\lambda$ the usual product measure on $2^{\lambda}$ can be
extended to a $\mathfrak{c}$\- additive measure defined on all subset of
$2^{\lambda}.$

Remark

Even the existence of a weakly compact cardinal and more generally the
reflection of all stationary subsets of $E^{\aleph_{2}}_{\aleph_{0}}$ would
imply as we know that there exists a model where $\aleph_{2}$ is inaccessible
in $L.$ So, this condition is inavoidable.

Remark

Let us remark the crucial fact that PFA itself implies the diamond on
$\aleph_{2}$, and the $2$-uniformization on $\aleph_{1}$ at the same time, the
reasoning is that PFA implies both non-existence of any form of both
$\aleph_{2}$ \- Kurepa tree and $\aleph_{2}$ \- Aronszajn tree - see [TJ03],
making $\aleph_{2}$ a weakly compact cardinal. Now, the made constatation
shows up that the proof of Tall in its paper [TF11], more precisely Theorem 17
is completely wrong - he based it on the wrong observation that PFA implies
that the square or (as he says the weak square on $\aleph_{1}$), even he cites
M. Magidor, in his as usual very deep result in the paper [MMAG01] when he
establishes that PFA implies the negation of weak square of double
$\aleph_{1}.$ Let us recall for the reader that the existence of a non-
reflecting stationary susbset of $E^{\aleph_{2}}_{\aleph_{0}}$ is equivalent
to such a combinatorial square-like principle. Even not mention that Magidor
as usual establishes the consistency power and the destruction of some
principles and hence the existence of very large cardinals as supercompacts
for claiming that the square ”has been left the game”, so PFA in particular as
we know it has a really strong consistency power issue…

Now, Let us cite the corresponding result from [MMAG01]:

Fact 2.6 ”(Magidor). PFA implies $\neg\ \square_{\kappa,\aleph_{1}},$
$\forall\kappa\geq\aleph_{1}$, whereas the statement: ”PFA
$+\square_{\kappa,\aleph_{2}}$ $\forall\kappa\geq\aleph_{2}$” is consistent,”
i.e ( ( PFA $\land\ \square_{\kappa,\aleph_{1}})=0,\ \
\forall\kappa\geq\aleph_{1}$) and ( ( PFA $\land\
\square_{\kappa,\aleph_{2}})\neq 0\ ,\ \ \forall\kappa\geq\aleph_{2}$ ), see
[MMAG01].

Corollary 1 - Fact 2.6

Under PFA $\mathcal{B!}$ fails in the part about $\aleph_{2},$ which implies
that it is not true the argument why $C(K_{A})$ is Weak Asplund, under (MA
$+\neg\ CH$ ), see [KK05] for the validity under (MA $+\neg\ CH$ ) of the
assertation: ($C(K_{A})$ is Weak Asplund, and $A$ has size $\aleph_{2}$).

Corollary 2 - Fact 2.6

Under PFA the principle (**) is not consistent, but since MA
($\sigma$-centered) holds then actually it is true that $C(K_{A})$ is in
Stegall‘s class and hence Weak Asplund, here $A$ stands for a set with size
$\aleph_{1}$ because.

Remark

PFA goes further than Corollary 1 - Fact 2.6 making the reasoning of [PW01],
see not valid as well, since by [NP92] - where the result [” There is a weak
Asplund, not weak ∗ fragmentable”] under ($V=L$) was firstly established and
used, we know that the following principle:

(**) There exists a coanalytic subset $A$ of the interval $[0,1]$, whose
cardinality is ($2^{\aleph_{0}}$) uncountable without any perfect subset.

(**) implies $CH$ and it is implied by the usual Godel Axiom of Constructible
Universum:

$V=L$

But PFA contradicts CH making it fails, in particular making (**) not true. As
we know by Namioka and years after that by [KK05] - (**) is even stronger than
the claim of [PW01], i.e [” There is a weak Asplund, not weak ∗ fragmentable”]
- (**) goes further it implies under some restrictions that [” There is a
compact in the Stegall‘s class, not weak ∗ fragmentable.”]

So Under PFA the negation of the statement [” There is a compact in the
Stegall‘s class, not weak ∗ fragmentable.”] is possible, since we can assume
that $\aleph_{1}$ is a measurable cardinal then the construction of the double
arrow space will lead us only to Weak Asplund Banach space - not weak ∗
fragmentable and the double arrow space will not be even in the Stegall‘s
class.

Corollary [PFA + there is a precipituous ideal on $\aleph_{1}$]  implies that
the statements:

  1. (1)

”Kalenda Compact $K_{A}$ is Weak Asplund, but not in the Stegall‘s class” is
true.

  2. (2)

”Kalenda Compact $K_{A}$ is in the Stegall‘s class, not weak ∗ fragmentable”
is false.

Here $A$ has cardinality $\aleph_{1}.$

Corollary PFA  implies that the statements:

  1. (1)

”Kalenda Compact $K_{A}$ is in the Stegall‘s class, not weak ∗ fragmentable”
is true if $A$ has cardinality $\aleph_{1}$

  2. (2)

”Kalenda Compact $K_{A}$ is Stegall‘s class, not weak ∗ fragmentable” is false
if $A$ has cardinality $\aleph_{2}$

Here $A$ has cardinality $\aleph_{1}.$

Remark

Even the existence of a weakly compact cardinal, more generally the reflection
of all stationary subsets of $E^{\aleph_{2}}_{\aleph_{0}}$ would imply as we
know that there exists a model where $\aleph_{2}$ is inaccessible in $L.$ So,
this condition is inavoidable, see [Jech]

Let us give the following corollary-remark,

Corollary-Remark 1.

Let $\aleph_{2}$ be an inaccessible cardinal, hence in $L$ as well, then it is
not weakly compact ( in $L$ as well) if there exists an atomemless
$\aleph_{2}$-complete, $\aleph_{2}$\- saturated ideal $I$ on $\aleph_{2}.$

Corollary-Remark 2.

Let $\kappa$ be a regular cardinal, i.e for instance $\aleph_{2}$, then any
$\kappa$-complete, $\kappa^{+}$\- saturated ideal $I$ on $\kappa$ is
precipituous, and in particular, if it exists - we can find a normal ideal
with these properties. $I_{NS}^{\kappa}$ is not $\kappa^{+}$\- saturated ideal
on $\kappa\geq\aleph_{2}.$

Corollary-Remark 3.

But it is consistent up to Woodin Cardinal that $I_{NS}^{\aleph_{1}}$ is
$\aleph_{2}$\- saturated ideal.

Corollary-Remark 4.

Let ( $\kappa$ be $\aleph_{2}$ or $\aleph_{3}$ ) if $I$ is an ideal on
$\kappa$ such that is normal $\kappa^{+}$-saturated, $\kappa$-complete ideal
on $\kappa$ then
$E^{\kappa}_{\omega}:=\\{\alpha<\kappa:cf(\alpha)=\omega\\}\in I.$

Corollary-Remark 5.

In particular, If Corollary-Remark 4. holds then if we assume that squares on
both $\aleph_{1}$ and $\aleph_{2}$ hold then there exists a stationary subset
$S$ of $E^{\kappa}_{\omega}:=\\{\alpha<\kappa:cf(\alpha)=\omega\\}\in I$ such
that any intersection with the ordinals below belongs to $I_{N}S$, more
precisely, this intersection belongs as well to the intersection: $I\cap
I_{NS}.$

Now, based on the previous discussion our main model has the following form:

New Form of Approximations and Consistency power

Definition ($\mathcal{AB}$) Let us assume that:

  1. (1)

Let $\mathcal{B}!!$ and Unif($\aleph_{1}$,$E$,$\aleph_{0}$,$\aleph_{0}$) and
Unif($\aleph_{2}$,$E$,$\aleph_{1}$,$\aleph_{1}$) hold. Henceforth, we know
that $\aleph_{2}$ does not carry any free $\sigma-$complete normal ultrafilter
and henceforth in particular is not weakly compact in L. Here $\mathcal{B}!!$
stands for the principle $\mathcal{B}$ with the modification that
$2^{\aleph_{0}}=2^{\aleph_{3}}=\aleph_{4}.$

  2. (2)

Let there exist a precipituous ideal $\mathcal{P}$ on $\aleph_{3}$. First
case, $\mathcal{P}=I^{\aleph_{3}}_{NS}$, then this is equiconsistent up to
existence of a cardinal of Mitchell order at least $\aleph_{4}$. More
precisely, by Gitik, see Jech, the consistrncy strenght is between
$\aleph_{4}$ and $\aleph_{5}.$

  3. (3)

$\aleph_{1}$ is not measurable cardinal in any transitive model of ZFC,
containing all ordinals. (and $\aleph_{2}$).

We obtain the following observation trivially from above:

Observation

Kalenda Compacts are in Stegall’s class with generating set $A$ with size less
or equal $\aleph_{2}$ iff we assume $\mathcal{AB}$ and in addition that
$\aleph_{3}$ carries a precipituous ideal, in particular, let the precipituous
ideal on $\aleph_{3}$ is the $I_{NS}^{\aleph_{3}}$.

And finally emanation by Shelah - we give the following:

Definition ($\mathcal{AB!!}$) Let us assume that:

  1. (1)

Let $\mathcal{B}!+$ Unif($\aleph_{1}$,$E$,$\aleph_{0}$,$\aleph_{0}$) (and
Unif($\aleph_{2}$,$E$,$\aleph_{1}$,$\aleph_{1}$)) hold.

  2. (2)

$\aleph_{1}$ is not measurable cardinal in any transitive model of ZFC,
containing all ordinals.

######  Proposition 0.1.

Assume one of the three principles: ($\mathcal{AB}$) or ($\mathcal{AB!}$)
($\mathcal{AB!!}$) then we have the following result obtained by Kalenda and
Kunen [KK05]:

  1. (1)

$(K_{A})$ is in the class $weak^{*}$-fragmentable spaces by some metric iff
$A$ is at most countable.

  2. (2)

$(K_{A})$ lies in the Stegall class iff $A\leq\aleph_{1}$. (Because satisfies
(*)).

###### Proof.

It suffices to establish that under the assumption ($\mathcal{AB}$) the
results in the paper of Kalenda and Kunen [KK05] hold.

Let $\mathcal{I}:={Id-Unif(S({\aleph_{2}}),\kappa,\mu)}$ stand for the proper,
normal, ideal on which the corresponding uniformization holds, see Shelah in
[SSh17] for the introduction and study of it. The notations are as follows
here: ”ladder” stands for the ladder system and the ”colour” - for the number
of the colours and $S_{\aleph_{2}}$ simply means the relativization of this
uniformization on the stationary subset $S_{\aleph_{2}}$ over $\aleph_{2}$ now
as Tall says in [TALL11] the existence of a non-reflecting stationary susbset
$S$ of the ” low ” stationary susbset of $\omega_{2},$ i.e
$E^{\aleph_{2}}_{\aleph_{0}},$ with a combination of the fact that
$S\subset\mathcal{I}$ by assumption, leads us to the conclusion that there is
a $Q$-set $A_{2}$ for the needed construction in the paper of [KK05]. Moreover
as we know such an existence of $A_{2}$ is equivalent to the existence of a
normal special ladder system, provided by ($\mathcal{AB}$). Here $A_{2}$ has
cardinality $\aleph_{2}.$ So, as we know in [KK05] $(K_{A})$ is weak Asplund.
Assuming that the second item of ($\mathcal{AB}$) does not hold, i.e
$\aleph_{2}$ does not carry a precipitous ideal and that it is not measurable
in any transitive model of ZFC containing all ordinals, we obtain as in [KK05]
that $(K_{A})$ is actually in Stegall‘s class, see more precisely [KS99].

For $\aleph_{1}$ the normal separable, first countable, Moore space and non-
metrizable is given by the uniformization, provided by ($\mathcal{AB}$).

∎

Open Question: If $I(U)=\mathcal{P}$ what is happening to the diamond and the
antidiamond principles actually?

As well, is it true that $I(U)$ contains $\mathcal{P}$ implies if $I(U)$ is
proper, then all $E\in P(\aleph_{2})/I_{NS}$, such that $E>0$ are reflecting.

Is it possible at all that $I(U)$ is proper assuming $\mathcal{P}.$

Question:

We give partly answer above as a consequences of an estableshed results by
Magidor in the case when PFA holds. PFA implies that $\aleph_{2}$ is weakly
compact cardinal, in particular, if $\aleph_{2}$ is measurable then it is
weakly compact in $L$, which implies the failure of the squares from below by
Magidor. So, if $\aleph_{2}$ is a measurable cardinal then we obtain a diamond
on that cardinal, so $I(U)$ is not proper.

Na památku strašlivé tragédie, která se stala na Filozofické fakultě /
Univerzity Karlovy / 21.12.2023, v Praze.

In memory of the terrible tragedy that occurred in the Faculty of Arts /
Charles University / 21.12.2023, Prague.

En recuerdo de la terrible tragedia ocurrida en la Facultad de los Artes / La
Universidad de Carlos Cuatro de Bohemia / 21.12.2023, Praga.

[KS99] Kalenda, Ondřej. ”Stegall compact spaces which are not fragmentable.”
Topology and its Applications 96.2 (1999): 121-132

[KK05] Kalenda, Ondřej, and Kenneth Kunen.”On subclasses of weak Asplund
spaces.” Proceedings of the American Mathematical Society 133.2 (2005):
425-429.

[TALL11] Tall, Franklin D. ”PFA (S)[S] and the Arhangel’skii-Tall problem.”
arXiv preprint arXiv:1104.2795 (2011).

[MMAG01] Cummings, James, Matthew Foreman, and Menachem Magidor. ”Squares,
scales and stationary reflection.” Journal of Mathematical Logic 1.01 (2001):
35-98.

[SSh17] Shelah, Saharon. Proper and improper forcing. Vol. 5. Cambridge
University Press, 2017.

[PW01] P. Kenderov, W. Moors and S. Sciffer, A weak Asplund space whose dual
is not weak* fragmentable, Proc. Amer. Math. Soc. 129 (12) (2001), 3741–3747.
MR 1860511 (2002h:5401

[NP92] I. Namioka and R. Pol, Mappings of Baire spaces into function spaces
and Kadec renorming, Israel J. Math. 78 (1992), 1–20. MR1194955

[TJ03] Jech, Thomas. Set theory: The third millennium edition, revised and
expanded. Springer Berlin Heidelberg, 2003.

[KKJV14] Kunen, Kenneth, and Jerry Vaughan, eds. Handbook of set-theoretic
topology. Elsevier, 2014.

[ShJT01] Shelah, Saharon, and Jan Trlifaj. ”Spectra of the $\Gamma$-invariant
of uniform modules.” Journal of Pure and Applied Algebra 162.2-3 (2001):
367-379.

[JT20] Trlifaj, Jan. ”The Dual Baer Criterion for non-perfect rings.” Forum
Mathematicum. Vol. 32. No. 3. De Gruyter, 2020.

[RGJT06] Göbel, Rüdiger, and Jan Trlifaj. Approximations and endomorphism
algebras of modules. Walter de Gruyter, 2006.

[KDSSh78] Devlin, Keith J., and Saharon Shelah. ”A weak version of $\lozenge$
which follows from $2^{\aleph_{0}}<2^{\aleph_{1}}$.” Israel Journal of
Mathematics 29.2-3 (1978): 239-247.


# Hamilton’s equations for relaxation function

Pierre Nazé<EMAIL_ADDRESS>Universidade Federal do Pará, Faculdade de
Física, ICEN, Av. Augusto Correa, 1, Guamá, 66075-110, Belém, Pará, Brazil

###### Abstract

The relaxation function is the cornerstone to perform calculations in weakly
driven processes. Properties that such a function should obey are already
established, but the difficulty in its calculation is still an issue to be
overcome. In this work, I proposed a new method to determine such a function
for thermally isolated systems, based on a Hamilton’s equations approach.
Observing that the microscopic relaxation function can be turned into a
canonical variable, one can choose the initial conditions of the solutions of
Hamilton’s equations to avoid the calculation of the average in the initial
canonical ensemble. The unbearable example of the quartic oscillator is solved
to corroborate the method. Extensions to the quantum realm and stochastic
thermodynamics are mandatory.

##  I Preliminaries

I start by defining notations and developing the main concepts to be used in
this work.

Consider a classical system with a Hamiltonian
$\mathcal{H}(\mbox{\boldmath$z$}(\mbox{\boldmath$z_{0}$},t)),\lambda(t))$,
where $\mbox{\boldmath$z$}(\mbox{\boldmath$z_{0}$},t)$ is a point in the phase
space $\Gamma$ evolved from the initial point $z_{0}$ until time $t$, with
$\lambda(t)$ being a time-dependent external parameter. Initially, the system
is in thermal equilibrium with a heat bath of temperature
$\beta\equiv{(k_{B}T)}^{-1}$, where $k_{B}$ is Boltzmann’s constant. After
that, the system is decoupled from the bath, and during a switching time
$\tau$, the external parameter is changed from $\lambda_{0}$ to
$\lambda_{0}+\delta\lambda$. The average work performed on the system during
this interval of time is

$\langle
W\rangle=\int_{0}^{\tau}\left\langle\partial_{\lambda}\mathcal{H}(t)\right\rangle_{0}\dot{\lambda}(t)dt,$
(1)

where $\partial_{\lambda}$ is the partial derivative in respect to $\lambda$
and the superscripted dot the total time derivative. The generalized force
$\left\langle\partial_{\lambda}\mathcal{H}\right\rangle_{0}$ is calculated
using the averaging $\langle\cdot\rangle_{0}$ over the initial canonical
ensemble. The external parameter can be expressed as

$\lambda(t)=\lambda_{0}+g(t)\delta\lambda,$ (2)

where, to satisfy the initial conditions of the external parameter, the
protocol $g(t)$ must satisfy the following boundary conditions

$g(0)=0,\quad g(\tau)=1.$ (3)

I consider as well that $g(t)\equiv g(t/\tau)$, which means that the intervals
of time are measured according to the switching time unit.

Linear-response theory aims to express average quantities until the first-
order of some perturbation parameter considering how this perturbation affects
the observable to be averaged and the process of average [1]. In our case, I
consider that the parameter does not considerably change during the process,
$|g(t)\delta\lambda/\lambda_{0}|\ll 1$, for all $t\in[0,\tau]$. In that
manner, using such a framework, the average work produced is [2]

$\begin{split}\langle
W\rangle=&\,\delta\lambda\left\langle\partial_{\lambda}\mathcal{H}\right\rangle_{0}-\frac{\delta\lambda^{2}}{2}(\Psi_{0}(0)-\left\langle\partial_{\lambda\lambda}^{2}\mathcal{H}\right\rangle_{0})\\\
&+\frac{\delta\lambda^{2}}{2}\int_{0}^{\tau}\int_{0}^{\tau}\Psi_{0}(t-t^{\prime})\dot{g}(t^{\prime})\dot{g}(t)dt^{\prime}dt.\end{split}$
(4)

where $\Psi_{0}(t)$ is the relaxation function, given by

$\Psi_{0}(t)=\beta\left\langle\partial_{\lambda}\mathcal{H}(0)\partial_{\lambda}\mathcal{H}(t)\right\rangle_{0}-\mathcal{C},$
(5)

where $C$ is a constant. Such a function is the cornerstone to perform the
calculations in weakly driven processes. Other features, such as optimal
protocols, depend almost exclusively on the relaxation function. Knowing how
to calculate $\Psi_{0}(t)$ is therefore mandatory.

##  II Constant $\mathcal{C}$

About the constant $C$, two options are possible: if

$\mathcal{C}=\beta\langle\partial_{\lambda}H\rangle_{0}^{2},$ (6)

then

$(\Delta
F)_{2}=\delta\lambda\left\langle\partial_{\lambda}\mathcal{H}\right\rangle_{0}-\frac{\delta\lambda^{2}}{2}(\Psi_{0}(0)-\left\langle\partial_{\lambda\lambda}^{2}\mathcal{H}\right\rangle_{0}),$
(7)

where $\Delta F$ is the difference of Helmholtz’s free energy between the
final and initial equilibrium states of the system. However, if

$\mathcal{C}=\beta\langle\partial_{\lambda}H\rangle_{0}^{2}+2(\langle W_{\rm
qs}\rangle-\Delta F)_{2},$ (8)

then

$(\langle W_{\rm
qs}\rangle)_{2}=\delta\lambda\left\langle\partial_{\lambda}\mathcal{H}\right\rangle_{0}-\frac{\delta\lambda^{2}}{2}(\Psi_{0}(0)-\left\langle\partial_{\lambda\lambda}^{2}\mathcal{H}\right\rangle_{0}),$
(9)

where $\langle W_{\rm qs}\rangle$ is the quasistatic work of the system.

##  III Usual method

The calculation of the relaxation function has some steps:

  * •

Calculate the solution of the non-perturbed Hamilton’s equations;

  * •

Take an average on the initial canonical ensemble.

If Hamilton’s equations have an explicit analytical solution, the problem is
manageable, although it becomes difficult to take the average if the initial
conditions are not linearly dependent on the solutions. In the case when it is
not, the problem becomes hard to calculate. Indeed, initial conditions must be
sampled according to the initial canonical ensemble, the equations numerically
solved with them, and the relaxation function becomes a numerical average over
them at the end. Observing that such a process holds only for one $\tau$, then
a whole analysis of the non-equilibrium dynamics becomes almost impossible to
obtain.

##  IV New method

The new method proposed has the advantage of not performing the average
anymore. To start, consider the following type of Hamiltonian

$\mathcal{H}(q(t),p(t),\lambda(t))=\frac{p(t)^{2}}{2}+V(q(t),\lambda(t))$ (10)

Under weakly driven processes, the effective Hamiltonian is the non-perturbed
one, given by

$\mathcal{H}_{0}(q_{0},p_{0},\lambda_{0})=\frac{p_{0}^{2}}{2}+V(q_{0},\lambda_{0}).$
(11)

Calling the microscopic relaxation function $\psi_{0}$ as a new canonical
variable

$\psi_{0}=\beta\partial_{\lambda_{0}}V(a,\lambda_{0})\partial_{\lambda_{0}}V(q_{0},\lambda_{0}),$
(12)

one needs to find its canonical conjugated $\theta_{0}$. Such a function can
be calculated by using the condition

$\\{\psi_{0},\theta_{0}\\}=1,$ (13)

where $\\{\cdot,\cdot\\}$ is the Poisson bracket on the canonical variables
$(q_{0},p_{0})$. Solving the equation, one has

$\theta_{0}=\frac{1}{\partial_{q_{0}}\psi_{0}}p_{0}+f(q_{0}),$ (14)

where $f(q_{0})$ is an arbitrary function. For simplicity, I consider
$f(q_{0})=0$. Solving Eqs. (12) and (14) to $q_{0}$ and $p_{0}$, one has

$q_{0}=G_{1}(\psi_{0},\theta_{0}),\quad p_{0}=G_{2}(\psi_{0},\theta_{0}).$
(15)

Substituting in the non-perturbed Hamiltonian, one has

$\mathcal{H}_{0}(\psi_{0},\theta_{0})=\frac{G_{2}(\psi_{0},\theta_{0})^{2}}{2}+\partial_{\lambda_{0}}V(G_{1}(\psi_{0},\theta_{0}),\lambda_{0}),$
(16)

whose Hamilton’s equations

$\dot{\psi_{0}}=\partial_{\theta_{0}}\mathcal{H}_{0},\quad\dot{\theta_{0}}=-\partial_{\psi_{0}}\mathcal{H}_{0},$
(17)

furnishes the solutions $\psi_{0}(a,t)$ and $\theta_{0}(a,t)$, which depend on
$\psi_{0}(0)$ and $\theta_{0}(0)$. In particular, the number $a$ is chosen to
satisfy

$\psi_{0}(0)=\beta\partial_{\lambda_{0}}V(a,\lambda_{0})^{2}.$ (18)

The relaxation function is expected to depend only on
$\langle\psi_{0}(0)\rangle_{0}$. In this manner, I use the following argument:
taking the average on the initial canonical ensemble in $\psi_{0}(t)$ is the
same thing as choosing $\psi_{0}(0)$ and $\theta_{0}(0)$ such that the
following restriction is achieved

$\langle\psi_{0}(t)\rangle_{0}=\langle\psi_{0}(0)\rangle_{0}(C(t)-\mathcal{C}).$
(19)

In general, this is achieved by choosing

$\psi_{0}(0)=\langle\psi_{0}(0)\rangle_{0},\quad\theta_{0}(0)=0.$ (20)

The relaxation function finally is

$\Psi_{0}(t)=\langle\psi_{0}(0)\rangle_{0}(C(t)-\mathcal{C}).$ (21)

Remark that by using such a method, the part of solving differential equations
remains, but taking the average in the initial canonical ensemble is avoided.
Also, the function $\langle\psi_{0}(0)\rangle_{0}$ has no relevance to the
general characteristic of the irreversible work and optimal control procedures
since it can be taken as a natural unit to work with. The constant
$\mathcal{C}$, according to the examples provided, is the one associated with
$\Delta F$. This result is not yet understood.

###  IV.1 Harmonic oscillator

Let us recover the relaxation function of the harmonic oscillator perturbed in
the stiffness parameter $\lambda(t)$. Consider its non-perturbed Hamiltonian

$\mathcal{H}_{0}(q_{0},p_{0},\lambda_{0})=\frac{p_{0}^{2}}{2}+\lambda_{0}\frac{q_{0}^{2}}{2}$
(22)

The microscopic relaxation function $\psi_{0}$ is

$\psi_{0}=\frac{\beta}{4}a^{2}q_{0}^{2}$ (23)

The canonical conjugate is

$\theta_{0}=\frac{2p_{0}}{\beta a^{2}q_{0}}$ (24)

Solving for $q_{0}$ and $p_{0}$, one has

$q_{0}=\frac{2}{\sqrt{\beta}a}\sqrt{\psi_{0}},\quad
p_{0}=\sqrt{\beta}a\theta_{0}\sqrt{\psi_{0}}.$ (25)

Substituting in the non-perturbed Hamiltonian, one has

$\mathcal{H}_{0}(\psi_{0},\theta_{0})=\left(\frac{\beta
a^{2}\theta_{0}^{2}}{2}+\frac{2\lambda_{0}}{\beta a^{2}}\right)\psi_{0}.$ (26)

The Hamilton’s equations are

$\dot{\psi_{0}}=\beta
a^{2}\theta_{0}\psi_{0},\quad\dot{\theta_{0}}=-\left(\frac{\beta
a^{2}\theta_{0}^{2}}{2}+\frac{2\lambda_{0}}{\beta a^{2}}\right).$ (27)

Using

$a=\left(\frac{4\psi_{0}(0)}{\beta}\right)^{1/4},\quad\psi(0)=\langle\psi_{0}(0)\rangle_{0},\quad\theta_{0}(0)=0,$
(28)

the solution is

$C(t)-\mathcal{C}=\cos(\sqrt{\lambda_{0}}t)^{2}.$ (29)

Adapting $\mathcal{C}$ to the quasistatic work, the relaxation function will
be

$\Psi_{0}(t)=\Psi_{0}(0)\cos{(2\sqrt{\lambda_{0}}t)}.$ (30)

which agrees with the relaxation function calculated in Ref. [3].

###  IV.2 Quartic oscillator

Consider the non-perturbed Hamiltonian of the quartic oscillator driven in the
stiffness parameter

$\mathcal{H}_{0}(q_{0},p_{0},\lambda_{0})=\frac{p_{0}^{2}}{2}+\lambda_{0}\frac{q_{0}^{4}}{4}.$
(31)

The microscopic relaxation function $\psi_{0}$ is

$\psi_{0}=\frac{\beta}{16}a^{4}q_{0}^{4}.$ (32)

The canonical conjugate is

$\theta_{0}=\frac{4p_{0}}{\beta a^{4}q_{0}^{3}}.$ (33)

Solving for $q_{0}$ and $p_{0}$, one has

$q_{0}=\frac{2}{\beta^{1/4}a}\psi_{0}^{1/4},\quad
p_{0}=2\beta^{1/4}a\theta_{0}\psi_{0}^{3/4}.$ (34)

Substituting in the non-perturbed Hamiltonian

$\mathcal{H}_{0}(\psi_{0},\theta_{0})=2\beta^{1/2}a^{2}\theta_{0}^{2}\psi_{0}^{3/2}+\frac{4\lambda_{0}}{\beta
a^{4}}\psi_{0}.$ (35)

Hamilton’s equations are

$\dot{\psi_{0}}=4\beta^{1/2}a^{2}\theta_{0}\psi_{0}^{3/2},\quad\dot{\theta_{0}}=-\left(3\beta^{1/2}a^{2}\theta_{0}^{2}\psi_{0}^{1/2}+\frac{4\lambda_{0}}{\beta
a^{4}}\right).$ (36)

Using,

$a=\left(\frac{16\psi_{0}(0)}{\beta}\right)^{1/8},\quad\psi_{0}(0)=\langle\psi_{0}(0)\rangle_{0},\quad\theta_{0}=0,$
(37)

and calculating $\mathcal{C}$ for the quasistatic work, the relaxation
function can be found numerically. Observe that the relaxation function is a
periodic function by the phase space configuration of the system. The
normalized relaxation function is depicted in Fig. 1. It is an even function,
which can be expanded into a cosine series with coefficients verified to be
non-negative until the 10th order. This indicates that its Fourier transform
is probably non-negative. Also, numerical calculation indicates that the
waiting time, given by

$\tau_{w}=\mathcal{L}_{s}\left[\frac{\Psi_{0}(t)}{\Psi_{0}(0)}\right](0),$
(38)

where $\mathcal{L}_{s}$ is the Laplace transform, is null for the same reason.
This is expected to thermally isolated systems [4]. Also, this indicates that
the system has shortcuts to adiabaticity for all $\tau$ [4]. In Fig. 2, the
excess work is numerically calculated for the linear protocol. The strong
inequality $\langle W\rangle\geq\langle W_{\rm qs}\rangle$ holds for it. The
excess work decreases to zero for large $\tau$, as expected.

Figure 1: Normalized relaxation function for a quartic oscillator. It was used
$\beta=1$ and $\lambda_{0}=1$. Figure 2: Excess work for a quartic oscillator.
It was used $\beta=1$, $\lambda_{0}=1$ and $\delta\lambda=0.1$.

## References

  * Kubo  _et al._ [2012] R. Kubo, M. Toda, and N. Hashitsume,  _Statistical physics II: nonequilibrium statistical mechanics_ , Vol. 31 (Springer Science & Business Media, 2012).
  * Nazé and Bonança [2020] P. Nazé and M. V. Bonança, Journal of Statistical Mechanics: Theory and Experiment 2020, 013206 (2020).
  * Acconcia and Bonança [2015] T. V. Acconcia and M. V. Bonança, Physical Review E 91, 042141 (2015).
  * Nazé [2023] P. Nazé, arXiv preprint arXiv:2305.17802 (2023).