# Rigorous definition of "Thermodynamic path"

We know the definition of thermodynamic state: A 3-tuple like (X,Y,Z) or (P,V,T) which we can assign to the system at a certain time. I also know "Thermodynamic path" refers to the path taken in the thermodynamic coordinates by the system from one state to another. But it seems not rigorous enough.

Can we introduce a more precise mathematical definition of "Thermodynamic path"? For example defining it as a unique mapping between two thermodynamic states?

P.S. I know physics is not math but I know a physical theory is a mathematical framework that fits best with our descriptions of nature. Classical thermodynamics as a theory is no exception.

• @AccidentalFourierTransform I know a possible path between two states is not unique in general. But every specific path is unique. Am I right? Nov 8, 2016 at 17:43
• I don't see what exactly you're lacking. The set of all thermodynamic states forms a space, and a thermodynamic path is just a path in the usual mathematical sense in that space. Nov 8, 2016 at 18:54
• @ACuriousMind Can I consider that "usual mathematical sense" a map or a function? Nov 8, 2016 at 18:57

If we consider a space spun by the thermodynamic variables $(p,V,T)$ (or some other work-like conjugated variables, like the magnetic moment $\mu$ and field $B$), then an equation of state may be given as the level set of a function (given by experimental data or some a priori considerations) defined on that space, i.e. $$F(p,V,T)=0$$ This level set viewpoint gives the explanation of why thermodynamical values are mutually dependant, i.e. why giving the value of two variables defines the third. Furthermore, that level set may be regarded as a 2-dimensional manifold $M$ embedded in the space of thermodynamic variables. Getting information about entropy or thermodynamic potentials may be viewed as a Legendre transform of that equation, but that's another story.

A (parametrised) thermodynamic path $\gamma(\lambda)\subset M$ can then be defined in the way you grasped in your question, as a curve on the 2-manifold $M$. In other words, let $\lambda\in\left[a,b\right]\subset \mathbb{R}$. Then we may define: $$\gamma:\left[a,b\right]\to M$$ $$\gamma(\lambda)=(p(\lambda),V(\lambda),T(\lambda))\in M$$ Note that one set of boundary values can define an infinite amount of curves.

For practical calculations in general physics we don't usually need to invoke so much rigour, so we just jumble around with differentials like apples until we get something satisfactory. In case of phase transitions, however, it becomes crucial to analyze the properties of discontinuities that arise in parts where the manifold is not smooth anymore. That, in part, is the work for which this year's Nobel in physics was given.

• Now that I've posted the answer I see that JalfredP answered it some time before me, and even invoked the isothermal hyperbole example. I'll let my answer stand anyway, in case more details are needed. Nov 8, 2016 at 19:04
• Since your answer was in a more formal manner, I accepted yours. But I voted up @JalfredP 's too. Thank you. P.S. This makes me think that analytical mechanics and thermodynamics are so similar in the way they interpret the phenomena. Right? Nov 8, 2016 at 19:20
• Thank you. Yes, in some ways they are similar, since in both cases we work with phase spaces of some sort. However, thermodynamics relies on it more heavily than classical mechanics, since the latter has some more independant formulations through Newton's equations, the principle of least action (Lagrangian mechanics), and D'Alembert's principle. In the end, of course they yield the same results, but the point is that we can sometimes avoid Hamiltonians and symplectic geometry, while in thermodynamics the formulations I know of use this formalism heavily. Nov 8, 2016 at 19:27
• This, of course, is an argument about thermodynamics as a phenomenological description and an autonomous discipline, without regard to its deeper explanation in statistical physics and molecular kinetic theory. Nov 8, 2016 at 19:29

For a closed system, the path of the process is determined not by what is happening inside the system but by what is happening at the boundary of the system with its surroundings. This can be specified by giving the temperature or the heat flux at the boundary (as a function of time), as well as the force per unit area, the volume, or the work done (again, as a function of time).

I really don't see what is wrong or non mathematical with your definition!

A path is a curve $p$ in parameters space $$p (s)=(P(s), V (s), T(s))$$ where $s$ is a parameter (maybe time) which describe how to get from one state to the other, exactly as you would do in phase space for classical mechanics. It is not unique as you can link two states in several ways.

As an example an isothermal hyperbole $P=Nk_B T/V$ linking two states is a path...