(Regarding isomorphisms: https://ilchimicoscettico.blogspot.com/2019/07/isomorfismi-caos.html)
What is a vector field? A space where a vector is defined at every point.
Put this way it seems abstruse and abstract, but each of us lives inside a vector space: we live within Earth's gravitational field. At every point where we move, a vector is defined that points toward the center of the planet - this is the gravitational force, which is why we walk and fall but don't float in the air. But at each point there is also associated potential energy: if you're on scaffolding 30 meters above ground, your potential energy is greater than that of someone standing on the street below you.
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| The gravitational field of the Earth-Moon system |
Like an electric field, the gravitational field is a conservative field. This means that the potential energy of a body is inherent to its position relative to the center of mass that causes the field. If we lift a body 10 meters and then bring it back to ground level, its potential energy at the end of the path remains what it was at the start. The potential energy of a body moving in a gravitational field depends solely on its position. That is, it's independent of the path taken from the initial to the final position. This can be described by the properties of the differential of potential energy as we have defined it: the differential of potential energy in a gravitational field is an exact differential.
In rough terms, let's say that a differential form is a function whose arguments are differentials (for differentials see here: https://ilchimicoscettico.blogspot.com/2019/01/dinamiche-equazioni-differenziali.html). For two variables x and y, the differential form ω will be ω(dx, dy) (actually, for a correct definition, much mathematics and many specifications regarding the space in which the differential form is defined would be needed).
In general, let φ be a closed path, regular and/or piecewise defined in an "appropriate space", and let ω be a differential form defined in the same space. If
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then the differential form is called exact.
Why the generalization? Because this happens in very different contexts, for example in thermodynamics. In the past I posted something about this (https://ilchimicoscettico.blogspot.com/2022/01/forme-differenziali-e-termodinamica.html), but it's good to expand the discussion. In that video you don't see integrals over a closed path, but it shows how dU is exact with another method: it shows that dU is closed.
Here too it's good to generalize. Let's take ω defined as above and as above let's admit it's a function of the differentials of two variables, x and y:
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If
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Then the differential form is called closed (it's the cross-derivative method). You can find a fairly exhaustive treatment here. For us now it's enough to say that closure is not a sufficient condition to define the exactness of a differential form, but if ω is closed and the space in which it's defined is a star-shaped set (in which every segment connecting a point to the center of the set is entirely contained in the set), then ω is also exact (and this can be demonstrated). The space in which a state function is defined has no "holes", so to speak, and that's why what's presented in the video holds up.
However, it's easier to talk about state functions using paths and particularly closed paths. Let's start with enthalpy, H, which is a measure of the heat content of a system. Hess's law tells us that the enthalpy difference between reactants and reaction products is the same whatever the intermediate products may be. Which is like saying that the enthalpy of the system depends on its initial and final states independently of the path taken to get from one to the other.
As for closed paths, let's take the best known one, the Carnot cycle (https://ilchimicoscettico.blogspot.com/2018/04/entropy-in-uk.html), and examine it looking at the system's entropy (S).
Let's take A as the starting point of the cycle. During the cycle, work was performed, heat was exchanged (so the entropy of the "universe" increased). Specifically, from the graph you can see that the system's entropy doesn't vary in the adiabatic sections of the path (where no heat is exchanged) while it varies in the isothermal sections (where the system is heated or cooled). But at the end of the cycle, the system's entropy remained the same. For the system, the integral of dS along the closed path equals 0.
So the state functions of thermodynamics are isomorphic to potential energy in a conservative field.
These will seem like banalities to many. To others they'll appear as incomprehensible stuff. But both groups should question one point: why don't these incomprehensible banalities find citizenship in the "narrative of science," AKA "science communication" or "popularization," while instead tons of nonsense have been written about S and the second law of thermodynamics, and just as much will be said and written? Perhaps we find the answer in that (despised) philosophy that was historically at the root of scientific thought: Francis Bacon's Novum Organum.
The idols and false notions that have penetrated the human intellect, fixing themselves deeply within it, not only besiege minds so as to make access to truth difficult, but even (once this access is given and granted) will rise again and be a cause of trouble even in the very establishment of the sciences: unless men, forewarned, arm themselves as much as possible against them. Four are the kinds of idols that besiege the human mind.
The idols of the tribe are founded on human nature itself and on the human tribe or race itself. Therefore it is falsely asserted that sense is the measure of things. On the contrary, all perceptions, whether of sense or mind, derive from analogy with man, not from analogy with the universe. The human intellect is like a mirror that reflects irregularly the rays of things, mixing its own nature with that of things, deforming and distorting them.
The idols of the cave are idols of man as an individual. Each person indeed (besides the aberrations proper to human nature in general) has a kind of individual cave or den that refracts and deforms the light of nature, either because of each person's own singular nature, or because of education and conversation with others, or from reading books and the authority of those who are honored and admired, or because of the diversity of impressions depending on whether they are received by a mind already conditioned and prejudiced or one that is clear and balanced. So the human spirit (as it appears in individuals) is so varied and greatly changeable and almost subject to chance. Therefore Heraclitus rightly affirmed that men seek the sciences in their small private worlds and not in the greater world common to all. (https://ilchimicoscettico.blogspot.com/2023/09/lontano-dallequilibrio-da-2500-anni.html, Editor's note)
There are then idols that derive almost from a contract and from the reciprocal relations of the human race: we call them idols of the marketplace because of the commerce and consortium of men. Men indeed associate through discourse, but names are imposed according to the understanding of the common people, and such erroneous and inappropriate imposition extraordinarily encumbers the intellect. On the other hand, the definitions or explanations with which learned men have provided themselves and with which they have protected themselves in certain cases have in no way served as a remedy. Rather, words do violence to the intellect and confuse everything and drag men into innumerable and vain controversies and fictions.
Finally, there are idols that have penetrated into the souls of men from various philosophical systems and from erroneous laws of demonstration. We call them idols of the theater because we consider all the philosophies that have been accepted and created as so many fables presented on stage.
And it must be said that several centuries later, everything continues to hold, even though today the idola theatri, the fables presented on stage, are emphasized.
The idola in Bacon are the pars destruens (destructive part). The pars construens (constructive part) consists of the tabulae: recording when the phenomenon occurs, when it doesn't occur, and the degree (quantity) with which it occurs: the roots of a quantitative and analytical approach. The approach that was used by Boyle, who owed much to Bacon (https://ilchimicoscettico.blogspot.com/2023/09/leggiti-kuhn-2x-by-starbuck.html).
And it's exactly this approach that has been completely lost for some time in public "scientific" discourse. An approach that is moreover incompatible with that mythology of "science" which, in fact, is the constitutive center of a new pseudo-religion, conveyed by a primary school catechism (https://www.sinistrainrete.info/societa/21518-andrea-zhok-credere-nella-scienza.html). And it seems useless to gesticulate to say that no scientific discipline is made to give human meaning to something (https://ilchimicoscettico.blogspot.com/2023/10/il-senso-e-la-sfiducia.html). But then again, today's protagonists of the idola theatri are those definable as neo-educated: trained in the ephemeral without any culture in the classical definition of the term, moreover profoundly hostile to the concept of classicism. But the very concept of scientific thought is classical. Elaboratable, debatable and all the rest, from what they tell me from Popper to Lakatos. While that training founded on the ephemeral is written on the sand of a shoreline: it vanishes into nothing and vanishing into nothing is good for every ephemeral season (read: the fashion of the moment). But it remains nothing, even if it's often widely glorified - in today, tomorrow it will disappear and prepare for a new and convenient ephemeral.
So much for the useless philosophy.
