Talk:Integrability conditions for differential systems

referred to in Caratheodory's infamous exposition on thermodynamics and measure theory

hi guys,

just annotating the talk page for now. i suspect when i'm done reading the paper i'll have more to say, but given my mentions of differential forms on earlier conversations (see Talk:Constant of integration), i feel somewhat relieved that i have some company.

On page 2^[1] of his famous exposition, we see that he mentions the Pfaffian, which lead me to this page.

sooooo relieved. maybe i'll have more to say when i'm done. not sure yet.174.3.155.181 (talk) 00:12, 11 July 2016 (UTC)[reply]

almost* done my first pass, but everything comes together around page 23. much of this exposition focused on exploiting what i'd call the "practical constants of integration", which is denoted with the zero subscript for the respective coordinate system being discussed.

translation by Delphinich is definitely better than nothing, but i took the first copy i could find. i had to correct some errors.

one error worth mentioning is the the second line of the two-line equation page 17, just above equation (28). it should be

d\epsilon _{2}+DA_{2}=N(y_{0},y_{1},\ldots ,y_{m})dy_{0}

, not

d\epsilon _{2}+DA_{2}=N(x_{0},x_{1},\ldots ,x_{m})dx_{0}

, because the first system's deformation coordinates are in terms of

x_{i}

and the second in terms of

y_{i}

edit (22:55ish GMT 12 July 2016): subscript for system of equations $G_{1},\ldots ,G_{\lambda }$ enumerated on page 5 is incorrect; correct form is $G_{0},\ldots ,G_{n}$

edit (21:39ish GMT 13 July 2016): subscript '0' missing for two-dimensional plane connecting $P_{i}$ with line G–i.e. $x_{0}=t$ (not $x=t$ ) 174.3.155.181 (talk) 21:43, 13 July 2016 (UTC)[reply]

far from done obviously, but it does seem the overarching theme puts an emphasis on basics of calculus. pretty cool once you see it come together, but i can see how those unfamiliar with calculus of variations don't respect it (which isn't a good thing). 174.3.155.181 (talk) 01:10, 12 July 2016 (UTC)[reply]

OOOOOO SH************, CARATHEODORY BUSTS OUT THE TERM "THERMODYNAMIC POTENTIAL" ON PAGE 26. MAD POTENTIAL THEORY FEELS HOLLA PAUL GARABEDIAN AWW YEAH 174.3.155.181 (talk) 01:46, 12 July 2016 (UTC)[reply]

k i have read the paper and, while i have a plan of attack in my head, i'm waiting for my mind's subconscious organisation algorithm to complete defragmentation of the most recently-added content.

at this time i want to share some quotes that, when i later add the relevant formulations from Caratheodory's work (and presumably tie it to other older ideas), are expected to improve readers understanding of how William Thomson, 1st Baron Kelvin's work aligns with the Fundamental theorem of calculus, also known as Stokes' theorem.
at the least, it is hoped the ensuing effort will help readers understand the existence of the hypothesised one-to-one correspondence between mathematical analysis and thermodynamics, where the latter was known to be the inspiration of the former during the times of Robert Boyle and probably his biggest fan, Sir Isaac Newton.
young, "modern" mathematicians may find it unusual that the root of the purely abstract concepts comprising the foundations of mathematics were once inspired by "Nature", but this is where a proper understanding of the fundamental theorem of calculus is crucial, as this deceptively simple result is powerful enough to summarise any experience, assuming the calculating machine has sufficient bit length (precision) to represent the rich, infinitesimal differential values

now to share the relevant (italicised) quotes, with commentary where appropriate.

1. "Now, although no other assumptions on the nature of heat can be made^[2], one can construct a theory that accounts for all of the results of experience..."
2. "One can derive the entire theory [of thermodynamics] without assuming the existence of a physical quantity that deviates from ordinary mechanical quantities, namely, heat."
3. "I have chosen an arrangement of the conclusions that differs from the classical proofs as little as possible, and likewise exhibits the parallelism that must necessarily exist between the main results of the theory and the picture that emerges from the measurements that are actually carried out.
4. "At the conclusion, I would like to draw attention to the fact that the notion of temperature is not included in the coordinates from the outset, but first appears as a result of certain equations of condition, which are presented on page 16. The grounds for which this conception of temperature is to be preferred are briefly hinted at in the final section; they originate in certain situations that give rise to radiative phenomena
5. "The second law that now comes into question is of a completely different nature: Namely, one has found that under all adiabatic changes of state that start from any given initial state certain final states are not attainable, and that such "unattainable" final states can be found in any neighbourhood of the initial state.
  However, since physical measurements cannot be absolutely precise this fact of experience includes more than the mathematical content of the aforementioned law, and we must demand that when a point is excluded, the same shall also be true of a small region around this point whose size depends upon the precision of the measurement..."
6. "Leaving from a given initial point, one can obviously actually arrive at any possible final form under the influence of particular external forces. However, one can do more: Namely, the change of form of the system S that takes place during an adiabatic change of state is a prescribed function of time. In other words: One can prescribe n functions
  $x_{1}(t),x_{2}(t),\ldots ,x_{n}(t)$
  and demand that the change of state that goes through them is of a sort that the timelike variation of the deformation coordinates $x_{1},\ldots ,x_{n}$ $x_{1},\ldots ,x_{n}$ will be represented by above sequence.
  - Carathéodory then mentions that the external work A is not a function of time, but state, and that the first law of thermodynamics can be written as
  $\int _{t_{0}}^{t_{1}}[d\epsilon +DA]=0$ where $DA=p_{1}dx_{1}+p_{2}dx_{2}+\ldots +p_{n}dx_{n}$ , and $p_{1},\ldots ,p_{n}$ are functions of $x_{0},x_{1},\ldots ,x_{n}$ ^[3] and $\epsilon =\epsilon _{1}+\ldots +\epsilon _{n}$ is the (total) internal energy of the system^[4]
  
  assuming we are dealing with quasi-static, adiabatic, change of state, which is defined as the difference between the limiting value of the external work and applied external work (will elaborate later, or try to).
7. Conversely any any curve in the space of $x_{i}$ that leaves $x_{0}$ constant can be regarded as the trace of the change of state of that sort; namely $x_{0}={\rm {const}}$ is equivalent to
  $d\epsilon +DA={\frac {\partial \epsilon }{\partial \xi _{0}}}d\xi _{0}+X_{1}dx_{1}+X_{2}dx_{2}+\ldots +X_{n}dx_{n}=0,$
  $X_{i}={\frac {\partial \epsilon }{\partial x_{i}}}+p_{i}$ ^[5]
  Now, Axiom II states there exists a state in an arbitrarily-chosen neighbourhood of an arbitrarily-chosen point that cannot be approximated by adiabatic changes of state.
  However the assumption of quasi-static, adiabatic change of state also means ${\frac {\partial \epsilon }{\partial \xi _{0}}}$ is not exactly zero.
  If the states that cannot be approximated (i.e. singular points that exist by Axiom 2) are omitted, then the above expression has a non-infinite and non-zero multiplier, denoted as 1/M, that bounds the expression for the first law
  $d\epsilon +DA=Mdx_{0}$
  Using this identity equation to replace the DA in $DA=p_{1}dx_{1}+p_{2}dx_{2}+\ldots +p_{n}dx_{n}$ , we obtain the relations:
  $Mdx_{0}=d\epsilon +DA={\frac {\partial \epsilon }{\partial x_{0}}}dx_{0}$
  $p_{i}=-{\frac {\partial \epsilon }{\partial x_{i}}}$
  which, when satisfied by the given coordinate system, result in a normalized coordinate system.
8. Let two simple systems S₁ and S₂ be given with normalised coordinates $x_{0},x_{1},\ldots ,x_{n}.$ , $y_{0},y_{1},\ldots ,y_{n}.$ [respectively]. These systems shall be separated by a fixed wall that is permeable only by heat... defined by the following properties:
  1. The deformation coordinates of the two systems in question can be varied independently of each other after the introduction of a coupling ^[6]
  2. Any arbitrary change of the form of the total system, when it is adiabatically isolated, equilibrium is reached after a finite time
  3. The total system S is then only found, but also always therefore in equilibrium, when a certain relation between the coordinates x_i, y_i of the form:
    $F(x_{0},x_{1},\ldots ,x_{n};y_{0},y_{1},\ldots ,y_{m})=0$
    is satisfied
  4. Whenever any two systems S₁ and S₂ are in equilibrium with a third system S₃ under analogous conditions, there likewise exists an equilibrium between S₁ and S₂
    This condition therefore means the same thing as saying that for the three equations:
    $F(x_{0},x_{1},\ldots ,x_{n};y_{0},y_{1},\ldots ,y_{m})=0,$
    $G(x_{0},x_{1},\ldots ,x_{n};z_{0},z_{1},\ldots ,z_{k})=0,$
    $H(y_{0},y_{1},\ldots ,y_{m};z_{0},z_{1},\ldots ,z_{k})=0,$
    which bring about equilibrium between S₁ and S₂, S₁ and S₃, S₁ and S₃. Each equation is a consequence of the other two.
    This is however possible only when the system of equations above are equivalent to a system of the form
    $\rho (x_{0},x_{1},\ldots ,x_{n})=\sigma (y_{0},y_{1},\ldots ,y_{m})=\tau (z_{0},z_{1},\ldots ,z_{k}).$
    In particular the condition from the equation in list item 3 (directly above) can then be replaced by two equations of the form:
    $\rho (x_{0},x_{1},\ldots ,x_{n})=\tau ,$
    $\sigma (y_{0},y_{1},\ldots ,y_{m})=\tau$
    in which τ means a new variable.
    One calls this quantity τ the temperature and equations ρ(x₀,...,x_n) and σ(y₀,...,y_m) the equations of state ^[7]
  - Section 7 Absolute Temperature relates two (normalised) systems S₁ and S₂ with equations of state
    
    ${\begin{aligned}d\epsilon _{1}+DA_{1}&=M(x_{0},x_{1},\ldots ,x_{n})dx_{0},\\+\;d\epsilon _{2}+DA_{2}&=N(y_{0},y_{1},\ldots ,y_{m})dy_{0},\\\hline d\epsilon +DA&=Mdx_{0}+Ndy_{0}\end{aligned}}$ ^[8]
  - Noting that both the parameters M,N and equations of state ρ(x₀,...,x_n), σ(y₀,...,y_m) of S₁ and S₂ are independent of each other, we introduce a change of variables to normalise the first law (expressed a sum of these systems):
    $du=\lambda [Mdx_{0}+Ndy_{0}]$
    where λ is a function of x₀,y₀,τ; M is a function of x₀,τ, and N is a function of y₀,τ.
    By the Leibniz's product rule, differentiating λN and λM with respect to τ gives:
    $M{\frac {\partial \lambda }{\partial \tau }}+\lambda {\frac {\partial M}{\partial \tau }}=0$ and $N{\frac {\partial \lambda }{\partial \tau }}+\lambda {\frac {\partial N}{\partial \tau }}=0$
    which gives a logarithmic derivative (with respect to τ):
    ${\frac {1}{\lambda }}{\frac {\partial \lambda }{\partial \tau }}$
    that can not depend on either x₀ or y₀^[9] thereby decomposing λ into a product of a function of x₀,y₀ and another function of a single variable τ:
    $\lambda ={\frac {\psi (x_{0},y_{0})}{f(\tau )}}$
    This expression allows further splitting of M and N such that they are a product of two factors: one depending upon τ, and the other upon x₀ (y₀ respectively):
    $M=Cf(\tau ){\frac {\alpha (x_{0})}{C}},\;\;\;\;N=Cf(\tau ){\frac {\beta (y_{0})}{C}}$
    where C is an arbitrary^[10] non-null constant.^[11]
9. To obtain the entropy coordinate as introduced by Rudolf Clausius, we first normalise the equations using the second last equation from list item 7, and the expressions for M obtained at the end of list item 8,
  $DA=-{\frac {\partial \epsilon }{\partial x_{1}}}dx_{1}-{\frac {\partial \epsilon }{\partial x_{2}}}dx_{2}-\ldots -{\frac {\partial \epsilon }{\partial x_{n}}}dx_{n}$ $DA=-{\frac {\partial \epsilon }{\partial x_{1}}}dx_{1}-{\frac {\partial \epsilon }{\partial x_{2}}}dx_{2}-\ldots -{\frac {\partial \epsilon }{\partial x_{n}}}dx_{n}$
  ${\frac {\partial \epsilon }{\partial x_{0}}}={\frac {t\alpha (x_{0})}{c}},$ ${\frac {\partial \epsilon }{\partial x_{0}}}={\frac {t\alpha (x_{0})}{c}},$ ^[12]
  and introduce a new coordinate η using an integral form of the expression for M obtained in list item 8:
  $\eta -\eta _{0}=\int _{a_{0}}^{x_{0}}{\frac {\alpha (x_{0})}{c}}dx_{0}$ $\eta -\eta _{0}=\int _{a_{0}}^{x_{0}}{\frac {\alpha (x_{0})}{c}}dx_{0}$ ^[13]
  which can be solved for x₀ since, by the last equation in list item 8, M, and consequently α are non-zero. The total differential of the energy function is now of the form
  $d\epsilon =td\eta -DA$ $d\epsilon =td\eta -DA$
  where η is called the entropy. Thus for systems S = S₁ + S₂ considered in list item 8, the total differential is:
  $d\epsilon =d\epsilon _{1}+d\epsilon _{2}=t(d\eta _{1}+d\eta _{2})-DA_{1}-DA_{2}.$ $d\epsilon =d\epsilon _{1}+d\epsilon _{2}=t(d\eta _{1}+d\eta _{2})-DA_{1}-DA_{2}.$
  and expressing their respective entropies η₁,η₂ with equations:
  $\eta _{1}+\eta _{2}=\eta ,$ $\eta _{1}+\eta _{2}=\eta ,$
  ${\frac {\partial \epsilon _{1}}{\partial \eta _{1}}}-{\frac {\partial \epsilon _{2}}{\partial \eta _{2}}}=0$ ${\frac {\partial \epsilon _{1}}{\partial \eta _{1}}}-{\frac {\partial \epsilon _{2}}{\partial \eta _{2}}}=0$
  as functions of x_i,y_k and the new η variables, the total differential again takes the form $d\epsilon =td\eta -DA$ .
  Carathéodory briefly mentions how physicists have overlooked the additive property of entropy in order to regard entropy as a physical quantity similar to the mass that is attached to any spatially extended body, which is probably among the most distinguishing features of his formulation compared to others.
  He lastly goes on to say that since the entropy only depends on x₀ in normalized coordinates, that it remains constant for any quasi-static adiabatic process, and any change of state of a simple system where the entropy remains constant is called reversible.
  we summarise the (brief) section on 'irreversible changes of state with two relevant passages:
  From this [the fact that the entropy of an isolated system never decreases], it follows that equilibrium will always come about when the entropy must decrease under all permissible virtual changes of state of a simple system, and it possesses a maximum in the equilibrium state.^[14]
  Any change of state under which the value of entropy changes is called irreversible.
  - Carathéodory proceeds to synthesise the above in a setting where we are given simple system S that may depend on the coordinates
    $\xi _{0},x_{1},x_{2},\ldots ,x_{n}$ $\xi _{0},x_{1},x_{2},\ldots ,x_{n}$
    and assume temperature scale is already determined.^[15] The following functions are determinable from the measurements:
    1. The equation of state S for any temperature scale τ.
      $\Phi (\xi _{0},x_{1},x_{2},\ldots ,x_{n})=\tau$
    2. The coefficients of the Pfaffian expression for the (external) work A:
      $DA=p_{1}dx_{1}+p_{2}dx_{2}+\ldots +p_{n}dx_{n}$
    3. The coefficients of the Pfaffian equation for quasi-static, adiabatic changes of state:
      $0={\frac {1}{\frac {\partial \epsilon }{\partial x_{0}}}}(d\epsilon +DA)=d\xi _{0}+X_{1}dx_{1}+X_{2}dx_{2}+\ldots +X_{n}dx_{n}.$ ^[16]
    to experimentally determine functions X_i, an approach similar to the definition of the derivative is used; that is, we consider changes in state of the system where only a single deformation coordinate, say x₁, increases by Δx₁, while x₂,...,x_n remain constant.
    We then measure the change Δξ₀ in ξ₀ that occurs during the process. When Δx₁ is sufficiently small, and the change of state occurs sufficiently slowly,^[17] we get:
    $X_{1}=-{\frac {\Delta \xi _{0}}{\Delta x_{1}}}$ ^[18]
    for the initial state. The Pfaffian in item 3 (of this subsection) must possess a multiplier λ that makes it a complete differential (i.e. satisfying the definition of the Pfaffian system on page 12) when quantities X_i satisfy certain differential equations that must necessarily exist for any actual system, as long as the theory is correct.
    
    I am going to stop here, as his remaining derivations employ techniques introduced in earlier sections, making the rest easier to follow.
10. One thus only needs to regard the state coordinates that we had up to now as functions of the position, and to modify the definition of the internal energy and the work done from the outside with the help of particular corresponding integrals
174.3.155.181 (talk) 00:54, 17 July 2016 (UTC) [reply]
References
1. ^ Carathéodory, Constantin (1909). "Untersuchungen ueber die Grundlagen der Thermodynamik" [Examination of the foundations of Thermodynamics] (PDF). Mathematische Annalen. 67. Translated by Delphinich, D. H.: 355–386.
2. ^ aside from assuming when two objects come into contact, heat flows from the object with higher temperature to the one with lower temperature
3. ^ Notice how $p_{1},\ldots ,p_{n}$ include $x_{0}$ , which is not a deformation coordinate? important to note as $x_{0}$ plays an important role as a "non-vanishing coefficient" in the Pfaffian.
4. ^ Caratheodory originally defines α as the total number of phases $\phi _{1},\ldots ,\phi _{\alpha }$ contained in system S; under the same change of variables (subscripts) that he employs earlier when setting up a simplex-like optimisation problem on page 5, it is okay to assume there are n phases.
5. ^ It may help viewing ξ₀ as a Augmented form (slack variable) that's introduced to facilitate the calculation of the "normalized" (bounded by M) Pfaffian.
6. ^ i think it's reasonably safe to suggest the introduction of a coupling between S₁ and S₂ implies connectedness (?) The coupling bit is interesting because, hypothetically, we could use a hyperboloid as a coupling, no? fix the points on the circumference of the circle corresponding to the respective space (i.e. circle C₁ for space S₁, etc) and connect them with a line (like the animation).
7. ^ here, i believe Carathéodory is implicitly exploiting the power of the real numbers. that is, if S₁ and S₂ are in equilibrium with eachother, then by def'n there is no net heat exchange. thus, an incredibly precise real value could presumably summarise this (shared) property between these two systems that may have radically different deformation coordinates (which, remember, summarise the phases of the respective system!).
8. ^ M and N bound S₁ and S₂, respectively, and their sum follows from the first law (sum of internal energy ε and external work A sum to near-zero).
9. ^ recall the Great Gauss Almighty's Theorema egregium expresses Gaussian curvature without depending on the surface, instead relying on measurements taken on the surface. it seems the logarithmic derivative for each system having the same form reflects this intuition(?)
10. ^ here, C is arbitrary because of the result from section 6, which showed the temperature scale is arbitrary by setting two equations of state equal to the same temperature, followed by demonstrating the equations that equal this temperature are not uniquely determined (page 16)
11. ^ decomposition seems to reflect the intuition from footnote 7(?) that is, temperature could be used to further separate the quantitative expression for each state (sort of like removing redundant information prior to comparing properties).
12. ^ Note: variable t is now the temperature
13. ^ Recall, on pages 12-13 that a₀,a₁,...,a_n are coordinates that are unreachable from the differential curves,and that make the Pfaffian equation a complete differential; the completeness presumably allows integration from a₀ to x₀
14. ^ this statement seemingly lends support to the Principle of maximum entropy, which can presumably be calculated using mathematical optimisation.
15. ^ Carathéodory assumes t is known to reduce the (already-lengthy) exposition; it follows from this assumption that both the internal energy ε and entropy η can be determined.
16. ^ On page 12, coefficients X_i of Pfaffian
  $dx_{0}+X_{1}dx_{1}+X_{2}dx_{2}+\ldots +X_{n}dx_{n}=0$
  are finite, continuous differentiable functions of x_i.
17. ^ "sufficiently slowly" to satisfy def'n of the quasi-static adiabatic change of state on page 10
18. ^ at the risk of belabouring "simple" definitions, recall that holding all deformation constants whilst varying one is the textbook definition of the partial derivative

[1] Carathéodory, Constantin (1909). "Untersuchungen ueber die Grundlagen der Thermodynamik" [Examination of the foundations of Thermodynamics] (PDF). Mathematische Annalen. 67. Translated by Delphinich, D. H.: 355–386.

[2] side from assuming when two objects come into contact, heat flows from the object with higher temperature to the one with lower temperature

[3] Notice how $p_{1},\ldots ,p_{n}$ include $x_{0}$ , which is not a deformation coordinate? important to note as $x_{0}$ plays an important role as a "non-vanishing coefficient" in the Pfaffian.

[4] Caratheodory originally defines α as the total number of phases $\phi _{1},\ldots ,\phi _{\alpha }$ contained in system S; under the same change of variables (subscripts) that he employs earlier when setting up a simplex-like optimisation problem on page 5, it is okay to assume there are n phases.

[5] It may help viewing ξ₀ as a Augmented form (slack variable) that's introduced to facilitate the calculation of the "normalized" (bounded by M) Pfaffian.

[6] think it's reasonably safe to suggest the introduction of a coupling between S₁ and S₂ implies connectedness (?) The coupling bit is interesting because, hypothetically, we could use a hyperboloid as a coupling, no? fix the points on the circumference of the circle corresponding to the respective space (i.e. circle C₁ for space S₁, etc) and connect them with a line (like the animation).

[7] re, i believe Carathéodory is implicitly exploiting the power of the real numbers. that is, if S₁ and S₂ are in equilibrium with eachother, then by def'n there is no net heat exchange. thus, an incredibly precise real value could presumably summarise this (shared) property between these two systems that may have radically different deformation coordinates (which, remember, summarise the phases of the respective system!).

[8] M and N bound S₁ and S₂, respectively, and their sum follows from the first law (sum of internal energy ε and external work A sum to near-zero).

[9] recall the Great Gauss Almighty's Theorema egregium expresses Gaussian curvature without depending on the surface, instead relying on measurements taken on the surface. it seems the logarithmic derivative for each system having the same form reflects this intuition(?)

[10] re, C is arbitrary because of the result from section 6, which showed the temperature scale is arbitrary by setting two equations of state equal to the same temperature, followed by demonstrating the equations that equal this temperature are not uniquely determined (page 16)

[11] sition seems to reflect the intuition from footnote 7(?) that is, temperature could be used to further separate the quantitative expression for each state (sort of like removing redundant information prior to comparing properties).

[12] Note: variable t is now the temperature

[13] Recall, on pages 12-13 that a₀,a₁,...,a_n are coordinates that are unreachable from the differential curves,and that make the Pfaffian equation a complete differential; the completeness presumably allows integration from a₀ to x₀

[14] this statement seemingly lends support to the Principle of maximum entropy, which can presumably be calculated using mathematical optimisation.

[15] Carathéodory assumes t is known to reduce the (already-lengthy) exposition; it follows from this assumption that both the internal energy ε and entropy η can be determined.

[16] On page 12, coefficients X_i of Pfaffian
$dx_{0}+X_{1}dx_{1}+X_{2}dx_{2}+\ldots +X_{n}dx_{n}=0$
are finite, continuous differentiable functions of x_i.

[17] "sufficiently slowly" to satisfy def'n of the quasi-static adiabatic change of state on page 10

[18] t the risk of belabouring "simple" definitions, recall that holding all deformation constants whilst varying one is the textbook definition of the partial derivative

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]