# Does distance exist in thermodynamic coordinates?

Consider a $$P$$-$$V$$ space. The question is a simple one: can we measure distance between two states $$(P_1,V_1)$$ and $$(P_2,V_2)$$?

Using simple Euclidian distance $$d^2 = (P_2-P_1)^2 + (V_2-V_1^2)$$ does not seem to work because the units of pressure are different than those of volume and these, or the squares of these, cannot be added owing to the difference in dimensions/units.

But we could divide the pressure and volume by some standard value and obtain dimensionless quantities.

Now, the pressure and volume, which are to be interpreted without any dimension or unit, could be used to measure distance between two thermodynamic states.

Does this imply anything relevant and is this approach valid? Are there other approaches to measuring distance between two thermodynamic states?

• Perhaps see 'Wasserstein metric' and related ideas, ex: arxiv.org/abs/1912.08405 Commented Dec 7, 2023 at 17:33
• I guess you might start by asking yourself what significance you would like to attribute to the distance in question. If you don't have anything particular in mind then any metric at all will serve, including the ones you propose to construct by choosing pairs of reference pressure and volume. But then, why bother? If you do have something in mind, then that should help in searching for something appropriate. Commented Dec 7, 2023 at 22:44

This is an old problem to find a natural metric for a thermodynamic process connecting two states. What you are suggesting does not really work because it is not natural in the sense that any other metric of the form $$k_p(p_1-p_2)^2+k_v(V_1-V_2)^2$$ would equally be justified.

The answer is not simple. One famous result that is originated in trying to improve on the works of Weinhold [4] is the so-called horse-carrot theorem of Salamon and Berry, for details see [1][2][3]; here I just quote it from the very easily readable [1]:

Theorem 1 (Discrete Horse-Carrot Inequality) In a process of $$K$$ near equilibrations of a simple system to states along a given quasi-static locus, $$K \gg 1$$, the entropy production is bounded below by

$$\Delta S_u \ge \frac{L^2}{2K} \tag{1}$$ Here $$L$$ is the thermodynamic length of the path $$L = \int \sqrt{-\sum_{i,j} \frac{\partial^2 S}{\partial X_i\partial X_j}dX_idX_j }\tag{2}$$ and the $$X_i$$ are a complete set of extensive variables of the thermodynamic system.

A closely related, continuous time version is given by

Theorem 2 (Horse-Carrot Inequality) In a sufficiently slow process in which a system traverses a given quasi-static locus, the entropy production is bounded below by $$\Delta S_u \ge L^2 \frac {\bar\epsilon}{\tau} \tag{3}$$ where $$L$$ is the thermodynamic length of the quasi-static locus in Eq. (2), $$\tau$$ is the duration of the process and $$\bar \epsilon$$ is a mean relaxation time.

[1] P. Salamon: Thermodynamic Geometry, Udine notes, https://salamon.sdsu.edu/

[2] J. Nulton, P. Salamon, B. Andresen, and Qi Anmin: “Quasistatic Processes as Step Equilibrations”, Journal of Chemical Physics, 83, 334-338, 1985.

[3] P. Salamon and R.S. Berry, : “Thermodynamic Length and Dissipated Availability”, Physical Review Letters, 51, 1127-1130, 1983.