# What are the applications of calculus of variations, if any, to the subject of thermodynamics?

If we apply calculus of variations to Newtonian mechanics, we can discuss of functionals such as the lagrangian and how optimizing it leads to the equations of motion. However, does there exist application of the subject in thermodynamics?

I'm not sure how to phrase it precisely but I think there is a similarity in the notions of the concept of path dependence and functionals. In the sense that, in both we are talking function of functions, for example the work:

$$W= \int_{V_1}^{V_2}P dV$$

Is a function dependent on the kind of path you take in the PV curve, this is extremely similar to the notion of a functional!

• Related: physics.stackexchange.com/q/51534/2451 and links therein. Jan 16, 2021 at 22:01
• Recommended reading: the article 'Making sense of the Legendre transform', which is about the interconversion between the lagrangian and the hamiltonian formulation of classical mechanics. The article is 11 pages; 3 pages are about 'the Legendre transform in statistical thermodynamics'. Jan 16, 2021 at 22:26
• @Qmechanic, I do not see a direct relation. The linked page and associated links deal with Statistical Mechanics. Here, the question is within a thermodynamic treatment. Jan 17, 2021 at 6:51

There have been many attempts to apply Lagrangean variational techniques in thermodynamics, the most famous one is probably Prigogine's minimum entropy production principle. A nice description of it is a series of publications by James Li [1,2,3,4] of which I summarize one here.

Starting with the result that the entropy $$S$$ in equilibrium is a convex function of its extensive parameters, the matrix $$[D^2 S]=\frac{\partial ^2 S}{\partial A_i \partial A_j}\le 0 \tag{1}\label{1}$$ is negative definite, where it is assumed that the extensive parameters $$A_i$$ represent conserved quantities, we wish to derive the non-equilibrium heat conduction (Fourier equation) from a variational principle on the steady state entropy production rate.

Now we assume that the equilibrium convexity property also holds for infinitesimal volumes and rates as follows: for any positive function of the coordinates $$f=f(x,y,z)>0$$ $$\int_V dV f\sum_{i,j}\frac{\partial ^2 S}{\partial A_i \partial A_j}\dot{A}_i\dot A_j \le 0 \tag{2}\label{2}$$

Li argues that since in equilibrium $$S$$ is a maximum of its variables there must exist $$f>0$$ such that the integral is a (total) time derivative, in other words, there must exist a time function $$\mathfrak K = K(t)$$ so-called thermokinetic potential that is derivable from the $$D^2S$$ matrix with an integrating factor $$f$$.

$$\delta \mathfrak K = \int_V dV f\sum_{i,j}\frac{\partial ^2 S}{\partial A_i \partial A_j}\dot{A}_i \delta A_j \tag{3}\label{3}$$

The thermokinetic potential can only decrease in time, at least in this near equilibrium, ie., linear regime, that is $$\delta \mathfrak K \le 0$$ and in steady-state it reaches its minimum and as such it can be used as variational principle when characterizing the steady state. This is not the minimum entropy production principle but rather the minimum thermokinetic potential production principle, a mouthful.

As an example take the heat conduction problem where we assume that the only extensive conserved parameter of interest is internal energy $$U$$, then $$S=S(T,U)$$ and ($$V=const$$ throughout): $$\frac{\partial S}{\partial U}=\frac{1}{T} \tag{4}\label{4}$$ $$\frac{\partial ^2 S }{\partial U^2} = -\frac{1}{T^2}\frac{\partial T}{\partial U}=-\frac{1}{c\rho T^2}\tag{5}\label{5}$$ $$\frac{\partial U}{\partial t}=c \rho \frac{\partial T}{\partial t}\tag{6}\label{6}$$.

If now one sets $$f=2T^2 >0$$ and assumes that the temperature is fixed at the boundaries by $$\delta T=0$$ then a thermokinetic potential can be defined as $$\mathfrak K= \int_V dV (\nabla T)^2 \tag{7}\label{7}$$ for an homogeneous isotropic thermally conducting body and thereby derive the Fourier conduction in steady state. By making various other choices for $$f$$ one gets different heat conduction equations, e.g., for the non-isotropic, inhomogeneous, temperature dependent conductivity cases, see for details in [1].

[1]"THERMOKINETIC ANALYSIS OF HEAT CONDUCTION", Int. J. Heat Mass Transfer. Vol. 7, pp. 1335-1339.

[2] Thermodynamics of nonequilibrium systems. The Principle of Macroscopic Separability and the Thermokinetic Potential, JOURNAL OF APPLIED PHYSICS VOLUME 33, NUMBER 2 FEBRUARY, 1962

[3]"Stable Steady State and the Thermokinetic Potential", THE JOURNAL OF CHEMICAL PHYSICS VOLUME 37, NUMBER 8 OCTOBER IS, 1962

[4] "CARATHEODORY’S PRINCIPLE AND THE THERMOKINETIC POTENTIAL IN IRREVERSIBLE THERMODYNAMICS", The Journal of Physical Chemistry 66.8 (1962): 1414-1420.

The change in entropy of a system between an initial thermodynamic equilibrium state and a final thermodynamic equilibrium state is an extremum. It is the maximum value of the integral of $$dq/T_{boundary}$$ over all possible process paths between the initial state and the final state, where $$T_{boundary}$$ is the temperature at the boundary (interface) of the system through which the heat dq flows.

• I am not sure that the maximum entropy principle is an example of the calculus of variations. For reversible processes isn't the change of entropy independent of the path? Jan 17, 2021 at 6:36
• @GiorgioP Well, for all process paths (both reversible and irreversible) between the initial and final states, the change in entropy is the same. But what I said was that the change in entropy of the system is the maximum value of $$\int_0^{\infty}{\frac{\dot{q}(t)}{T_{boundary}(t)}dt}$$ over all possible paths, where $\dot{q}(t)$ is the rate of heat flow at time t, and $T_{boundary}(t)$ is the temperature at the interface between the system and surroundings at time t during the process. The integral has multiple maxima (all the same value for all the infinite number of reversible paths). Jan 17, 2021 at 13:56
• I agree, but only for reversible path $\dot q(t)$ can be written in terms of thermodynamic variables describing the system. In the general case, I do not see a way to have a functional of system's variables; or do you mean it as a functional of $q(t)$? Jan 17, 2021 at 15:01
• @GeorgioP I'm only talking about the heat flow rate and the temperature at the boundary of the system, not the internal thermodynamic variables. These are certainly defined for both reversible and irreverile paths. So I do mean that it is a functional of $\dot{q}(t)/T_{Boundary}(t)$ Jan 17, 2021 at 15:07
• No, I am emphasizing that. I consider Clausius' inequality as an upper bound for $\int dq/T_{boundary}$. In this sense, it establishes a variational principle for this integral which is maximum, and equal to the entropy variation, for a reversible path. However, I do not see how to transform it into something where calculus of variations can be used to obtain a path. A path in thermodynamic space, in my opinion, is precluded by the fact that there are irreversible transformations not corresponding to quasi-static processes, therefore not described by thermodynamic variables of the system. ... Jan 17, 2021 at 15:24