Stochastic and Macroscopic Thermodynamics of Strongly Coupled Systems

We develop a thermodynamic framework that describes a classical system of interest $\mathcal{S}$ that is strongly coupled to its thermal environment $\mathcal{E}$. Within this framework, seven key thermodynamic quantities—internal energy, entropy, volume, enthalpy, Gibbs free energy, heat, and work—are defined microscopically. These quantities obey thermodynamic relations including both the first and second law, and they satisfy nonequilibrium fluctuation theorems. We additionally impose a macroscopic consistency condition: When $\mathcal{S}$ is large, the quantities defined within our framework scale up to their macroscopic counterparts. By satisfying this condition, we demonstrate that a unifying framework can be developed, which encompasses both stochastic thermodynamics at one end, and macroscopic thermodynamics at the other. A central element in our approach is a thermodynamic definition of the volume of the system of interest, which converges to the usual geometric definition when $\mathcal{S}$ is large. We also sketch an alternative framework that satisfies the same consistency conditions. The dynamics of the system and environment are modeled using Hamilton’s equations in the full phase space.

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Popular Summary

The molecular world contains many examples of tiny machines and behaviors that mirror their larger, human-scale counterparts. Molecular motors, built from proteins and fueled by chemical energy, shuttle cargo around the interiors of cells. Strands of RNA, a molecule essential to carrying out genetic instructions, can be stretched and contracted like rubber bands. These microscopic systems seem to obey the laws of thermodynamics, the study of how heat and other forms of energy flow within a system as well as between a system and its surroundings. But thermodynamics traditionally applies only to large, macroscopic systems. We have developed a theoretical framework for describing thermodynamics at both the molecular level and at macroscopic scales.

Our theory focuses on properly accounting for the strong interaction between a microscopic system and its thermal surroundings, which blurs the boundary between the two. Key to that accounting is a thermodynamic definition of volume. Rather than measuring the amount of space (which is hard to describe for microscopic systems that do not have clearly defined surfaces), our definition is based on a system’s effect on its surroundings. We find that this definition of volume lets us precisely define key thermodynamic properties for a microscopic system. Quantities such as internal energy, work, and entropy in our framework obey the first and second laws of thermodynamics at both microscopic and macroscopic scales.

Future work could relax some of our assumptions and encompass systems that exchange molecules with their environment, such as a protein that collects and discards water molecules over time.

Figure 1

Schematic depiction of single-molecule setup (not to scale). A molecule of RNA attached to two micron-size beads is immersed in aqueous solution and manipulated using optical tweezers. The work parameter $\lambda$ denotes the distance from the fixed micropipette to the center of the laser trap.

Figure 2

Isothermal-isobaric setup for the RNA molecule in aqueous solution. The container is closed off by a frictionless piston whose vertical height $h$ is treated as one of the many ($\sim 10^{23}$) degrees of freedom of the environment $\mathcal{E}$.

Figure 3

The reversible work required to insert a pebble into a bucket of water, at constant pressure, is $P{V}_{\mathrm{peb}}$.