Stochastic thermodynamics, a new area of physics, aims to better understand and interpret thermodynamic concepts that are not in equilibrium. These fields have made it possible to understand the various thermodynamic processes that operate in finite time better over the years.

Mike DeWeese and Adam Frim, both researchers at the University of California Berkeley (UC Berkeley), recently conducted a theoretical study that explored the entire space of thermodynamic cycles, including the changing bath temperature. The results were presented in a paper published by Physical Review Letters. They used geometric methods. Thermodynamic geometry can be used to understand the thermodynamic system’s response by studying the control space.

DeWeese explained to that “For example, for a gas inside a piston,” one coordinate in the space of control could correspond with the experimentally controlled volume and another to temperature. If an experimentalist turned those knobs, it would plot a trajectory in this thermodynamic area. The thermodynamic geometry assigns each curve a “thermodynamic length” that corresponds to the maximum dissipated energy for a given path.

Researchers can use thermodynamic geometry to explore interesting research questions such as how to manipulate a given system of nanoscopic particles, erase some information or create a quantum or classical heat engine.

DeWeese stated that the main purpose of this paper was to find the most efficient way for a microscopic engine to produce the most work for the fuel it uses. While most thermodynamic geometry studies have focused on optimizing control of systems with certain initial and final settings in the past, we were more interested in creating optimal closed curves that could be used as heat engines.

Over a century ago, the main rules for how large engines can be run efficiently, such as those in cars, were first described. This was before the laws of thermodynamics were formulated. Frim and DeWeese expanded these theories to be applicable to microscopic engines that operate in finite time. These engines, unlike slow-operating large engines, are not in thermal equilibrium. They are heavily affected by the thermal fluctuations in their surrounding environment.

DeWeese stated, “In any undergraduate thermodynamics class, we learn that pressure vs. volume for a gas in an cylinder, and consider any closed cycles that return to the same point, the area within the curve gives us the amount of work you get from this heat engine. This assumes that the gas in the cylindrical moves slowly so that it remains in equilibrium with the outside world. This P-V diagram can be analogized to microscopic engines by a small particle attached at a spring that is subject to thermal fluctuations.

Frim and DeWeese discovered that the P-V diagram of a microscopic engine could be used to plot cycles. However, Frim & DeWeese were able to see that one cycle provided enough useful work. The length of the closed curve was also found to correlate with the amount of “useful labor” lost to dissipation, i.e. heating up the environment but not going towards the engine’s operation.

DeWeese stated that “our result is relatively straightforward conceptually.” In thermodynamic geometry, length is something like dissipation. We then considered the following: If you consider a closed loop (a cycle) that has a fixed perimeter, what is the area within this loop? It turns out that it is the cycle’s work output. Therefore, optimal cycles should have low output and high dissipation.

The researchers used classical geometric results to find optimal protocols to set a limit on all closed cycles’ efficiency. This could be a significant contribution to the development and design of efficient microscopic heat engine designs. This team of researchers established a general limit on the efficiency irreversible thermodynamic cycle efficiency. Its implications could extend far beyond the microscopic engines they considered.

DeWeese stated that one of their long-term goals was to develop the theory necessary for engineers to design and construct small, efficient engines. This could be a significant area in nanotechnology. We are also motivated to study the function and structure of molecular motors as well as other molecular’machines, which we can see in all living things.

De Weese & Frim speculate that natural evolution may have chosen efficient molecular machinery in their research. The rules they discovered could help us predict the structure and functions of molecular machines, which are so common in biology.

DeWeese said that isoperimetric inequalities, which are the interplay between lengths and areas closed curves in geometric approaches to Physics, could have many implications for the future. Our mathematical bound is much more realistic than the previous results, which assumed that the engine was in thermal equilibrium with the environment (or heat tub) at all times. However, we still assume that the system’s control parameters are being changed slowly and that it is being driven slowly. We now want to extend our results to systems further from equilibrium.