0
$\begingroup$

In thermodynamics, and more precisely when talking about continuous systems, some sources [1, 2] introduce functionals of state: $$F[s(x), \dots]:=\int_VdV(x)f(s(x),\dots,x)$$ In order to derive continuity equations for each extensive quantity. Also in the Enthalpy wikipedia page one finds the notation $H(S[p],p,\{N_i\})$ which suggests a functional.

I imagine that for this latter case this is done to emphasize that the entropy is a functional that is extremized with constraints at the thermodynamical equilibrium, so that is one link with the calculus of variations.

For the continuum case, is there a practical point in making this link ? Does the formal derivation of continuity equations rely on the calculus of variations?

EDIT: The implication of the existence (or not) of a variational principle is discussed in other linked questions. I am more particularly interested in whether or not it is necessary in some situation to distinguish between the functional derivative: $$\frac{\delta S[p]}{\delta p}$$ and the usual partial derivative $(\partial S/\partial p)$

References (in French):

[1] J. P. Ansermet, Thermodynamique, Chapter 10

[2] Stuckelberg, Thermocinétique phénoménologique galiléenne

Improve this question
$\endgroup$
4
  • $\begingroup$ The Gibbs free energy can be seen a functional ought to be minimized via the calculus of variations. See this question physics.stackexchange.com/q/797774 $\endgroup$
    – User198
    Commented May 18 at 13:30
  • $\begingroup$ I do not know the Ansermet book but Stuckelberg assumes local equilibrium so that the thermodynamic system under consideration can be broken up into sufficiently small but thermodynamically homogeneous pieces. It is assumed that these pieces can be described with macroscopic parameters that would also describe the whole system were it be in equilibrium. The integration replacing the finite summation of the quantities just expresses the assumption that the small enough pieces can be taken as homogeneous. No Volterra-style functional analysis is involved. $\endgroup$
    – hyportnex
    Commented May 18 at 13:45
  • $\begingroup$ Thank you for your comments! The linked question is interesting in its own right. I guess a precision on my question is whether the notion of functional derivative is necessary/useful in thermodynamics. Probably in the continuum there is a link between the functional derivative and the eulerian/lagrangian views but am unsure of its precise meaning. For discrete thermodynamics system probably functional derivatives are not useful ? $\endgroup$
    – GvPStack
    Commented May 18 at 15:22
  • $\begingroup$ Setting up a variational calculus for thermodynamic processes that is similar to the Euler-Lagrange formalism has been an unfulfilled dream for 150 years. The closest to that may be Bronsted's formalism that is very much in the spirit of the Lagrange-D'Alembert principle of virtual work of constrained virtual displacements but, in general, it is not a minimum (maximum) principle and it cannot be because it would be contrary to experiment. It "works" only close to equilibrium, but far from equilibrium it can lead to instabilities or even oscillations, see, e.g., Belousov-Zhabotinskiy. $\endgroup$
    – hyportnex
    Commented May 18 at 18:03

0

Reset to default